The game starts with each person being dealt two cards; one of those cards is face up and the other one is face down. There are two ways that the initial betting is done, depending on the place you choose to play 5 card stud. It is either done through an ante that everyone has to pay before the game starts, a bring-in bet that is done by the person that has the lowest card dealt to them or through some combination of the two. Either way, these are bets that are forced. The person immediately to the left of the person that is either the dealer or else the person with the forced bring-in bet has the action on the first round (referred to as second street), with the option given to them to raise, call or fold.
Once the first betting round has resolved itself, another card is dealt face up to each person. This is Third Street. During this round, the person that bets is the person that has the highest two-card hand showing on their board. They have the option of checking or betting. If they choose to check, then the next person has the same options. These same options are available until someone chooses to bet instead of check, at which point the next person then has the option of calling, raising or folding.
Once the betting round is over, one more card is dealt face up (Fourth Street) with the same betting options available. Once that betting round is over, Fifth Street is dealt face down to each player and a final round of betting commences. If there are still two players left in the pot after the final round of betting, each player shows their hand and the best five card hand takes the pot.
The reason that five card stud as a game became so popular so early on in the history of poker is that it provided a very nice compliment to the original game of poker; five card draw. In the five card draw game, you could not see any of your opponent's cards and therefore the only thing you had to go on was their body language and betting. In five card stud however, you can see at least three of the five cards that your opponent has (and in some cases you can actually see four of the five cards as some variants do not deal the last card face down) and therefore this combined with the same betting and body language information you can get from draw poker allows you to make a much more educated guess as to what your opponent might have.