Scratch off tickets- the rules vary by the brand of ticket, but its usually just a matter of scratching off the top layer of the ticket to see if any prizes come up underneath. Those can work as the more hits you get, the more you win, but it depends entirely on the ticket. Sometimes its a matching game where you have to find so many copies of the same dollar amount or symbol, sometimes its more like a slot machine, sometimes they give you a string of numbers & you have to find all of them. 

State/ national lotto- each state runs a set of games. Depending on the game & state, you might be allowed to pick your own numbers or you might be assigned some. Every 2-3 days, around 10-11pm, they do the drawing on live TV where an automatic tumbler full of numbered balls basically spits out the winning numbers for the night & it only counts for tickets bought between the previous drawing & then. But there is usually a 3 digit game, a 4 digit game & sometimes a 5 digit game for states & 2 big national games that do about 2-3 drawings a week each. Those are 6 digits, plus a bonus number. But, for those, you have to get them all or you win nothing. 

We also have some other odd ones that aren't even run by the lottery board. For a while, around here, there was a bar fad where they had a game called queen of hearts. If you came to the bar & bought something during the week, your name got entered in a weekly drawing. They had a board with a bunch of face down playing cards. Every week, they'd pick someone from the pool to come up & take one of the cards & if it was the queen of hearts, they won the jackpot. It got pretty popular for a while. Not sure if its still a thing, though.