The message of this song is very difficult to understand if you are not familiar with Christian theology and belief. Even many so-called Christians will have problems understanding these concepts because they use many metaphors from the Bible, so I'll try to give a lot of explanations to make it easier to understand, but I recommend to read under the "Meaning" tab before reading these explanations.

The first three lines of this song (spoken) are part of the Lord's Prayer, the most important prayer for Christians (see complete prayer with explanations here)

THY= (old English possessive form) Your.
In old English the second person singular and plural were different:
Singular: Thou – thee – thy – thine
Plural: Ye – you – your – yours
Compare: We – us – our – ours
We often use the old singular forms when we speak to God in prayers.

THY KINGDOM COME= May your kingdom come. Here the verb "come" is a subjunctive form expressing something unreal, in this case, a wish, not a fact. Present subjunctive forms take no –s for the third person singular.
KINGDOM= The Kingdom of God is the central theme of Jesus' message. It has a double dimension. On the one hand it is similar to heaven, the spiritual state where God reigns in perfect harmony and happiness and where good people will live after death. On the other hand the Kingdom of God is a society of justice and peace here on earth. Jesus wanted people to be good to go to heaven, but he also wanted people to be good to make this world a better and peaceful place. He wasn't killed for preaching about heaven, he was killed for preaching about justice and equality here on earth.

IN THE PURSUIT= Looking for, struggling for.

FABULOUS= Extremely pleasing or successful.

AN ACTIVIST= A person who fights for an idea, who takes action to make things change (for example Greenpeace activists who fight for protecting nature), but also, in a negative sense, a warrior, someone who fights and uses violence (even kills) to get his objectives. This paragraph in the song is negative and reflects the sinful nature of man. We all want to be happy (we're all in the pursuit of happiness) but we look for happiness in the wrong place, in earthly pleasures and material things (a life so fabulous), and the sentence "so we fight for what's ours like an activist" is talking about how selfish and materialistic we are (this is mine and I won't share it, and I will fight anyone who threatens my possessions)

STRUGGLE= Fight, violence.

'CROSS THE ATLAS= All over the world (across the Atlas). "Atlas" are a range of mountains in Morocco, but it's also a collection of maps from all over the world. So the song says that everywhere people are fighting for material possessions and success, looking for happiness in those things.

WHEN ADAM GAVE UP HIS OWN PALACE= When Adam lost his Paradise. The Jewish and Christian Bible tells a story to explain that humans may be born good, but have a tendency towards evil (are not perfect). In that story God created Adam and Eve, the first humans, in a Paradise on condition that they would not eat the fruit from a forbidden tree hidden in the middle of the garden. They disobeyed God, and so they were expelled from Paradise. Before that, they were perfectly happy, after that, outside Paradise, we have to struggle to be happy, and we can never find perfect happiness for long.
In this sentence GIVE UP means "to lose or abandon" and "palace" is a metaphor for Paradise.

RIGHTFUL= Having a proper claim over something. The rightful owner is the true owner. The world's rightful owner is Jesus/God.

THE SECOND ADAM= Jesus is said to be the second Adam, because the first Adam brought sin into this world, and Jesus brought salvation to this world.

AT HAND= Very close, very near; available, possible to get.

DIED FOR THE SINS OF MAN= (missing subject: Jesus) According to Christian theology, Jesus died for the sins of man (a sin is an evil action). Sins separate God from men because God is perfect and sins are imperfections, so an imperfect man cannot get in full contact with a perfect God. But when Jesus died, he broke that separation, the fault was compensated and men and God could be reunited again (a very simple explanation, I know, but...)
MAN is a generic word used for humanity, all human beings, men and women. This was the normal usage in the past, but today some people prefer to say "humans" or "men and women".

BRING US BACK= Take us to the original place. The particle "back" often means "return to the origin":
- Put the book back on the table (it was on the table, you took it, now put it on the table again)
- Please, come back to our house whenever you want (you are in our house, you can come to our house again in the future)

HIS ORIGINAL PLAN= God created man to be with them in harmony. Sin broke that harmony. Jesus' death restored that harmony, so his death brought us to God's original plan (at least Jesus opened the door so those who wanted to be in harmony with God could do it, nobody is forced to cross that door though).

WHERE WE RULE OVER THE LAND= To rule over something is to govern that thing, to have power over that thing. The complete sentence is "his original plan where we rule over the land...", which means that God's original plan was that man ruled over the land... In the Bible, when God gives a man or people power over something, it is never to exploit it, but to take good care of it.

MEET EVERY DEMAND= If you meet a demand, you react correctly to every need or request (you do things right).

FOR US KINGS= We can use US or WE together with a noun to say that we are part of that group:
- We Russians are used to the cold (if I say "we are used to the cold" I don't give any information about what group of people I'm taking about. If I say "Russians are used to the cold" I don't say that I am also a Russian and so I am used to the cold too).
- This book is specially written for us men  (it's written for men, and I'm a man)
We can use the plural pronoun YOU in the same way:
- Oh, you Italians are so elegant!
- I'm going to buy something for you friends.

TAKE A STAND= To publicly express an opinion about something, especially to say whether you support or are against something. To confirm your ideas and act accordingly.

THE KINGDOM IS IN US= Jesus often said that the Kingdom of God is inside us, in our hearts. He is then referring to the earthly kingdom, the society of peace and justice. So the sentence "the kingdom is in us so we choose when to advance" means that the power to work for justice and peace is inside us, so we can choose when to act on it and work to bring a better society.

YOU GOT= (coll.) You've got; You have.
"You got the keys inside this kingdom" means, in this kingdom we Christians are trying to create, you have an important role (figuratively we say that you have the keys for something when something depends on you, when you have the power to do it). So it's a call for action. Don't wish for a better world and wait for other people to do it, you have the key to do it, so come on and move, do something to change the world now!

LIFT UP YOUR HEAD= (lift = raise, elevate) Be proud; feel powerful (to lower your head is to feel humiliated or powerless). But in this context it could also mean look up to the sky and find inspiration in God. (I think the first meaning is more probably though)

IT'S ONLY BEGUN= This is only the beginning, nothing is going to stop us now and our work for a better world will grow and grow.

KEEP HOLDING ON= Continue waiting, don't despair.
"Keep holding on and you'll see the Son" (= Jesus, the Son of God). This sentence means: be confident, we know that Jesus is going to come back and he will bring the Kingdom of God to perfection on earth, so all your efforts won't be in vain. Fight for a better world and wait happily because one day Jesus will come back and make all your efforts worthy. The Second Coming of Jesus has (like the kingdom) a double meaning. He will come back to earth one day, but if you die before that, you will meet him in heaven, so either way, if you are good, "you'll see the Son".

EVERYBODY SING IT NOW= The verb "sing" has no –s because it is not a 3rd person singular, it is an imperative form. The imperative form has no subject, but it can have subject in two occasions, "You" to emphasize (especially to insult or estate your power) and "Everybody/Somebody/Anybody" to refer to one or more unknown people (impersonal):
- Sit down, please (no subject)
- You shut up! (a strong command)
- Somebody help me! (I need help, I don't care who helps me)
- Everybody go out (I want all the people in this room to leave)

YOU WOULD DIE= The verb "would" is used in reported speech as the past form of "will" (I heard you say that you would die for me = you said: "I will die for you"). And the verb "will" may express future but more often it expresses volition (desire, what you want), so "I will die for you" may be interpreted as "I want to die for you", "I am willing to die for you", I want to do it.

CAUSE= (coll.) Because.

THE KEYS TO LIFE= The chance or the way to be happy and have a good and full life.

IN MY PRIDE I HAD THEM STOLEN= "Pride" is a bad thing, it's the feeling of being superior and better than the rest. So Jesus gave you the keys to life (the instructions to be good and have a good life), but because I was proud, vain, I lost those "keys", I lost the possibility of having a good and happy life (so pride stole my keys to life, "in my pride, I had those keys stolen").

REDEEM US= Save us from our sins and imperfections, make us spiritually free.

THE KEYS CAN NEVER BE STOLEN, THEY'RE JUST MISPLACED= Remember that the keys refers to the instructions to be good and happy. If you are a bad person you lose the keys and you can't enter life/paradise. But we must remember that when Jesus died for us, he gave us the keys forever. If we fall back into sin, we don't lose the keys, they are just misplaced (when you misplace something you put it in the wrong place and you can't remember where they are), but not lost, so we always have the chance to turn back to Jesus commandments (instructions) and recover the keys to life.

WE RISE FROM THE ASHES= If a house burns down, it is destroyed, reduced to ashes. If we are reduced to ashes, our life is destroyed (because we had a bad life and did it all wrong). But if you turn back to Jesus (listen to him again), you can "rise from the ashes", that is, you can be a good person again and mend your life.

STAND FOR THE MASSES= Defend the people (probably referring to the weak and the poor, who can't defend themselves)

PROCLAIM= Say firmly and publicly.

GIVE UP= Surrender, stop trying.

IT'S NEVER ENOUGH /ɪnʌf/ = It's not enough. This "Never" is simply an emphatic negative form.

SIT BACK= The complete expression is "to sit back and do nothing".

HE HAS A PLAN= It refers to God. Christians believe that God lets people be free to do good or wrong (we call that "free will"), but we also believe that God is Lord of history and he has a general plan for humanity, and that plan is positive. So when times are really bad and we think that humanity is moving towards chaos, we must have faith and hope that He (God) has a plan for us, so in the end, everything will be alright.
"It's never enough for us to sit back and hope He has a plan" = You should never relax and do nothing an simply hope that God will make a better world. No, you have to work for a better world yourself, because God transforms history using people like you and me, not using a magic wand to change everything. God's plan is to make a better world using people like you, so don't sit back and wait for God to act, stand up and do something now!

WE GOTTA= (coll.) We've got to, we must.

STAND UP= Fight for your rights or for justice; defend your cause.

FACE THE COLD= Confront bad times and difficulties; fight for a better world even when everything is against us.

BOLDNESS= Bravery, courage.

AIN'T= (coll.) The negative form of the verb to be and to have:
- I ain't tired, I can keep working.
- I ain't got no money left, can you give me $5?
- We ain't never givin' up the keys= We're never going to abandon the keys (give them away)

YOU GAVE= "You" refers to God/Jesus. He promised us a better future and a paradise (here on earth, some day, or after we die, in another dimension).
So you gave us wonderful promises, but we didn't believed you and "walked away", abandoned you and the right path.

THE PRICE YOU PAID= The price that Jesus paid to let people come back to God was death. He died on the cross to save us and reconcile us with God again (explaining why his death saved us would be too long for these little notes).
"But now I see the price you paid was enough for us to live victorious" = I was often pessimistic and/or too lazy to act, but now I realized that what you did for us was enough to save us, so we can live a wonderful life (if we choose so) and have victory over sin and evil.