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For example, if someone attacks with a 3/3 creature, and you have a 5/5 creature that will ultimately block and destroy his creature.
If they boosts their creature with a +3/+3 spell, can you counter the spell? If they boost their creature with an equipment card, during his combat attack phase, can you counter that?
What, if anything, does this have to do with the "Stack"?
It depends on what he's doing, and what you have available.
If your opponent has a card like Serra Avenger, which is a 3/3, and he attempts to cast Giant Growth, giving the avenger +3/+3 until end of turn, then you can counter that with a spell like Cancel or by activating the ability of Judge's Familiar (assuming he can't or chooses not to pay the cost associated with the familiar). Otherwise, the aveneger, now a 6/6, will swallow a 5/5 in combat.
Note that he can cast the Giant Growth during his attackers step, or during your blockers step (after you've declared blockers). Consider the following scenario:
Him: delcare attacks
Him: declare Aveneger as attacker
You: block with Thundermaw Hellkite, a 5/5
Him: cast Giant Growth on Avenenger
Result: Aveneger, 6/6 lives, has 5 damage on it, and Hellkite is dead
It's important to note that this is not equipment, but an instant speed spell. Equipment plays by different rules: here's how those scenarios might play out.
If your opponent has a card like Manaforce Mace then he can't equip that during his combat phase, because he can only attach it at sorcery speed. But if you had three basic land types, then you'd have +3/+3 allowing you to take out the Hellkite in the above scenario. Note that this wouldn't really be a surprise like the Giant Growth would be, because you'd have to do this during combat.
If your opponent has a card like Cranial Plating then he could equip that during his combat phase with the activated ability. You can counter this only if you have something that allows you to counter abilities (not spells), for example Stifle
Another option would be something that allows equipment at instant speed, like Brass Squire. You could do this:
Him: delcare attacks
Him: declare Aveneger as attacker
You: block with Thundermaw Hellkite, a 5/5
Him: tap Brass Squire, attaching Manaforce Mace (which was already on the field, not from your hand, and assuming you have 3 basic land types) to Avenger.
Result: Aveneger, 6/6 lives, has 5 damage on it, and Hellkite is dead
116.3c If a player has priority when he or she casts a spell, activates an ability, or takes a special action, that player receives priority afterward. Of course, you have a chance to respond to any or all of those, and logically, it's to your advantage to pass priority after each one (otherwise you give your opponent more information than necessary), but it's not required.
using the term 'equipment card' in your question is confusing. Is your opponent using an instant/sorcery spell to provide the +3/+3 or an actual equipment artifact?
Because of this confusion, I will answer this question as if your opponent boosted his creature before attacking. (same stack rules will apply, but in almost all cases your opponent will not be able to play socercys or use equipt abilities [other abilities that boost power and toughness are still available])
If the opponent is casting an instant/sorcery spell to provide the +3/+3 the answer is YES, you can play a counterspell and then block and destroy his creature.
If the opponent is boosting his creature by use the 'equipt' ability of an artifact he controls (or any ability on a permanent), NO, you cannot play a counterspell. You can however play a different spell that would destroy or remove the artifact from his creature and then block.
Before I answer, I'm not sure that you understand what a spell is, so I'm going to cover that first. My apologies if you know.
A spell is a card on a stack, which is to say a card that has been cast but hasn't yet resolved, hasn't been countered and hasn't otherwise left the stack.
e.g. If you cast Llanowar Elves, it is a spell from the time you take it out of your hand until it enters the battlefield.
e.g. If you cast Giant Growth, it is a spell from the time you take it out of your hand until it resolves to give a +3/+3 bonus to a creature.
A continuous effect is not a spell.
e.g. The +3/+3 granted by Giant Growth is not a spell.
Cards on the battlefield are not spells.
e.g. An artifact attached to a creature is not spell.
Now on to the questions.
If they boosts their creature with a +3/+3 spell, can you counter the spell?
Depends on the spell you used to counter.
If they boost their creature with an equipment card, during his combat attack phase, can you counter that?
It's not clear what you mean.
If they are casting an Equipment spell, you can counter that. They wouldn't normally be allowed to do cast an Equipment spell in the Combat Phase, though.
If they are using the Equip ability Equipment, you can counter that, there are a very few cards (e.g. Stifle) that can do it. They wouldn't normally be allowed to use Equip in the Combat Phase, though.
If they are attaching the Equipment using an ability other than Equip (e.g. Brass Squire's), Stifle would do the trick again.
Keep in mind that you can't counter permanents (objects on the battlefield) and the effects of their static abilities (e.g. Bladed Bracers's +1/+1), just abilities on the stack and spells.
