BRING TEARS TO YOUR EYES= If something bring tears to your eyes, you cry/weep.

THE ONE YOU SHOULD CALL WAS STANDING HERE ALL ALONG= The person you need when you feel down is me, I’ve always been here for you.
ALL ALONG= All the time, always.

I WILL TAKE YOU IN MY ARMS= I will hold you, I will embrace you, I will put my arms around you.

HOLD= Embrace, hug, put my arms around you.

RIGHT= exactly. A word we use to emphasize time and place (I want it right here, right now!)

WHERE YOU BELONG= The place where you feel at home.
If you belong to a person or thing, they own you:
- This car belongs to John = It’s John’s car
- That screw belong to the kitchen door.
If you belong in a place or a group, that is where you feel good and accepted.
- Sorry, I don’t belong here = I don’t feel well in this place or with you guys.
- Now that you married my sister, you belong in this family = you are part of our family
- I belong in California = That’s the place where I feel good, where I feel at home (because I that’s my home or simply because I love the way I feel when I’m there)
- You belong in a mental home = You should be locked up in a mental institution because you are crazy.

TILL= (also ‘till) Until.

MY LIFE IS TRHOUGH= I die.
If something is through, it has finished.

THIS I PROMISE YOU= I promise you this.
Placing the direct object at the beginning of the sentence is not “correct” grammar, but it is very poetical and more beautiful.

LIFETIMES= A lifetime is the period of time a person is alive, from the cradle to the grave. If you say “I’ve loved you forever in my lifetime” it means “I’ve always loved you, since I was born”. But here he goes much more emphatic and says “in lifetimes before”, which means (using the idea of reincarnation) that I have lived many lives before this one, and in each one of them I have always loved you (that is really quite a thing to say, right?)

...NEVER / WILL YOU HURT ANYMORE= Again, they are changing the correct order of the sentence to make it sound more poetical. The correct order would be: “And I promise you (that) you will never hurt anymore”. HURT= suffer, feel pain. ANYMORE= Never again.
HURT can be used transitively or intransitively:
- She's hurting me = She’s making me suffer (either emotionally or physically)
- I stepped on broken glass and, gosh! It hurt! (it caused me pain)
- Stop squeezing my hand, you’re hurting me = you are causing me pain
- Stop squeezing my hand, I’m hurting = I’m feeling pain
- When she left me, It hurt so much = her departure caused me a lot of pain
- I know I’m hurting now, but it will soon pass = Now I’m suffering, but soon I’ll be ok

I GIVE YOU MY WORD= I promise you.

I GIVE YOU MY HEART= I love you and always will.

VOW= /vaʊ/ A very solemn promise.

FOREVER HAS JUST BEGUN= In this sentence, the word “Forever” is not an adverbial of time, as usual, but a noun, the subject of the verb begin. The idea is similar to this: “the future has just begun”.
When he says “and with this vow, forever has just begun” (beautiful expression), he means “I really promise you I will love you forever”, so with this vow I’m making, that forever officially begin.

LOVING DAY= A loving day is a day full of love.

WON’T GO AWAY= Will not disappear.

OVER AND OVER I FALL= Again, the normal order of the sentence has been changed for poetical reasons. It should be “I fall over and over”. There are two possible interpretations to this:
1- I fall very often, again and again, repeatedly, frequently.
The complete expression is “over and over again” but we can just say “over and over” for short.
2- Another possibility is that this repetition is just for emphasis, so the idea is, “I fall over”, and to make it more emphatic he says “I fall over and over”. FALL OVER means to trip, lose your balance and fall down (metaphorically in this case), so he says that “without you in my life, I fall over continually”