All of these games are designed to play well in a large group. They aim to encourage players to talk and have a fun time in a large group.
Say Anything (3-8 Players)
When I describe Say Anything to people who've never played it, I say it's Apples to Apples with a lot more freedom. One person asks a question such as "where would I go on my dream vacation?" Everyone then writes down an answer. After everyone has written one, the "judge" picks their favorite in secret. Then players have to place their tokens on which one they think the judge picked. If they're right, they get points, if not, they don't. Whoever has the most points after some number of rounds (there are official guidelines in the rules, but really, it can be open ended so long as people are having fun) wins.
Say Anything gets people talking about the questions. It can be a great way to get to know people. I've heard of it being used at youth groups or small groups to get to know fellow members. It could also be used in an office to get to know coworkers, or as a team building exercise.
There is a risk that some questions can go to risqué answers, but that all depends on who's playing. There is a family version that attempts to eliminate the remote possibility of such answers.
Wits and Wagers (3-20 Players) Well the box says 20, I'd say more depending on how you want to run teams.
W&W is a trivia game, but don't run away screaming just yet. The game is about knowing things, but not exactly. It helps to have a general sense of things, but some questions are so obscure, no one will get it right. Actually, most games will see no one write down the exact answer. You see, W&W is all about numbers. In a round, a question is asked. Everyone writes down a number. Then, the answers are ordered from low to high. Players place their token on the answer they think is the closest to the right one without going over. So it doesn't matter if you don't know the exact answer, but if you know an estimate, you can do well. At worst, go for the insane bet, and see if it pays off.
There have been 3 versions of this game, Wits and Wagers, Wits and Wagers Family, and Wits and Wagers Party. I like them all. They're all functionally the same game, just different questions, and some different means of scoring points. I was a playtester for Wits and Wagers Party, and I did receive a free copy of the game. This in no way affects what I think of the game, since I already liked the system before I got a free game.
Wits and Wagers has been hit or miss for me. Some people love it, and some people dislike it. They point to the lack of player interaction. While I see what they're saying, a party game is meant to be the lubricant that gets the conversation started.
Mafia/Werewolf (6-20ish Players)
This is an experience game. There are two sides in the game, the Mafia or Werewolves, and the normal people. Each side is trying to eliminate the other. The normal people generally outnumber the other side 2:1 or 3:1, but the Mafia/Werewolves know who each other are. The game is divided into 2 phases. In the first phase, the M/W group awakens and kills off one of the normals. Then everyone wakes up to find someone missing/dead. They then kill off someone in an attempt to extract justice for the previous murder. At this point the game sounds pretty violent, but it really isn't. There's some strategy in who you kill off and when as a Werewolf, and as a normal, you're looking for any odd behavior to give you a clue as to who is working against you.
This game is a lot more fun with a great storyteller moderating things.
Dixit (3-12 players, depending on the set. Odyssey goes to 12, regular Dixit only to 6)
Dixit is similar to Apples to Apples. Players have a hand of cards, except instead of words, they have artwork. The artwork is simply fantastic. There is great depth to each card, and so much variety. Each card has these little nuances that players can key on, and there really is no wrong way to describe these cards. I'd love to have posters with various Dixit cards, that's how amazing the art is.
Party games are all about what you think will be the most fun for the people involved.
If they love storytelling, it's hard to beat Mafia.
If they're creative artistic types, I say Dixit.
Have a bunch of trivia fans? Go for Wits and Wagers.
Say Anything really is the easiest game to get people to play. It's accessible, there's not much other knowledge required, and it's a great way to get to know people.
On a budget, I say get the Family version, and then learn how to play Werewolf with a deck of cards. None of these games are bad, I enjoy all of them for different reasons. Say Anything is probably the best bet to not fall flat. That's something you really want when buying games on a budget.