First off, make sure not to call a "bet" a "raise". If you can check, that is you aren't facing an amount you have to call, then when you put in chips it is called a bet. If you have to put in some amount of chips to continue with the hand, and you want to increase the pot, it's called a raise. If it is confusing, just remember this old poker adage: "You can't raise yourself."
In No-Limit Hold'em, the minimum bet is usually the big blind. The minimum raise is going to be the amount of the previous bet or raise called. For example, in a 1/2 nl game, the minimum you can raise before the flop is going to be to make $4... you are calling the $2 blind and then raising $2. If you make it four, the next player to act can raise to a minimum of $6... remember you are only raising $2.
This can get tricky when a player is all-in. Keep in mind that a player can always push his whole stack into the middle in NL, even if his stack is less than what a min-bet or min-raise would be. The question that arises often is whether a shortstack shove reopens the action to an original better.
For example, let's say you are playing NL holden with blinds at 100-200. You are first to act after the flop and you bet 500. A shortstacked player then shoves all-in for 700. Two other players call behind him. Now it's back up to you and you want to reraise. This scenario leads to arguments all of the time and I've seen lots of floor staff get it wrong.
Rules are going to be different from place to place, but under the TDA rules, it opens up the action again to you if the shortstacked player's raise is at least 1/2 the minimum. So in the example I gave, the shortstacked player would have to have made it 750 for you to be able to reraise when it got back to you. Other places don't use the 1/2 minimum rule and won't let the action be opened back up unless the full minimum amount is met.
Here is Rule 41 of the TDA :
41: Raises A: A raise must be at least the size of the largest previous bet or raise of the current betting round. If a player raises 50% or more of the previous bet but less than the minimum raise, he must make a full raise. The raise will be exactly the minimum raise allowed.
B: In no-limit and pot limit, an all-in wager of less than a full raise does not reopen the betting to a player who has already acted and is not facing at least a full raise when the action returns to him. In limit, at least 50% of a full raise is required to re-open betting for players who have already acted. See Illustration Addendum.