Poker probabilities can be done naively even at the tables, just counting the cards and doing the percentages in your head. I still find this kind of fun and sort of effective, even if I will show you some useful shorthand methods below.
What I mean by naively is simply to divide the cards you believe will make you win the pot by the total cards you have not seen. Look at this example turn situation, where you are last to act with a bet before you:
If you get an ace or a nine here, you are winning (or splitting) the pot. There are four aces and four nines not accounted for, and 46 unknown cards. Thus, you have 8/46 or a bit over 17% chance of making your hand.
A shorthand way of computing this is to count the cards, or outs, you have to make your hand. Each out accounts for a roughly 2% chance of making the hand. In this case the "outs shorthand" would indicate 16%, which is probably close enough in most cases. If you are guaranteed to see two cards, for instance when your opponent is all-in on the flop, you can use 4% (which is simply 2% twice, or a rough shorthand for 1/47+1/46, which is about 4,3%).
The term usually used in poker for this "chance of winning" is pot equity. With that in place, we are ready to look at two other basic concepts, namely pot odds and later implied odds.