You are way too generic, so it is hard to give a precise answer...
So the best answer is: it depends!

1. This one is easy: someObject.aClassVariable; or define getters, ie. methods that just return a value from the class (but they are less used in Processing).
2. If the variable holds a primitive value, from boolean to int, from float to char, you can just use == to compare them.
If it is an object, from String to anything you made yourself, you must use the .equals() method, but for your objects, you must define this function yourself inside your class.

Well it's not that easy. I think, i'd better share my code with you.

" Well it's not that easy"
Well, yes, it is that easy.
If you are talking about your score function, you have to pass the object(s) as parameter, like:
void score(Atom a1, Atom a2, int templ)
{
  if(a1.ALyloc == a1.ARyloc && a1.ALyloc == mouseY && skipScore == 0 && a1.ALcolor == a1.ARcolor){
  // ...
}

Thanks for a quick response! Almost understood. I guess I got how to rewrite my score function, but what do I do in draw function? 

Ah, the way it was formatted (no indentation, a separator before it), I thought score() was outside Atom, at same level than draw().
Quick advice: use Ctrl+T (or Edit > Auto Format) to format your code, so that such confusion can be avoided.

If I understood the purpose of score(), perhaps it is better outside of the class. Or not, it depends...
If you need to compare two Atoms, or the current Atom to another one, you need to pass them as parameter.
Let's keep it in Atom, somehow it can be simpler.
Then you pass one Atom parameter:
void score(Atom atom, int templ)
{
  // ...
}
You must compare each atom against each other, so you need a nested loop:

1.       for(i = 0; i<atom.length - 1; i++){
2.   atom[i].collider();
3.   // We start at i+1 to avoid comparing everything twice, and to avoid comparing an atom to itself
   
4.   for (j = i+1; j < atom.length; j++) {
   
5.     atom[i].score(atom[j],i);
6.   }
   
7.   }

(formatting can help here too...)

So you're trying to compare next two atoms by adding 1 to j? I don't know if you have tried  this code yourself, but they still do not animate like they're supposed to. The purpose of score is comparing colors of atoms and setting the score, so I think it's better to have this function outside Atom class (that's why I asked how to access variables from a class).

" So you're trying to compare next two atoms by adding 1 to j?"
No, I compare atom i with all the others (coming after this one, actually).

You pass a single atom, so don't try to index it, since it isn't an array. Since you call score() with atom[i] parameter, the index was already used.
In short: replace atom[i] with atom in your function.

Well, it still doesn't work. I don't think I'm doing something wrong, but could you test the original code and fix it.