Cocktail is an arcade game form-factor. A cocktail machine is a dedicated cabinet that is in the shape of a small table. The table itself will have a glass top, and the monitor will be mounted underneath the glass (facing upwards). You sit down at them rather than stand (like a normal arcade game).
This style of game was most popular in the 1979 to 1983 arcade game era, with most game produced during that time available in a cocktail version. Very few cocktail games were made after the mid 80s, but older cocktails are often converted to newer games (which is why you may sometimes see a NEO GEO or even a Street Fighter 2 cocktail).
Gambling machines (most often Video Poker), will often have a modified cocktail form. That is, they will be installed directly into the surface of a bar (without having their own separate table). This is so the casino can make extra money from customers who are drinking at the bar.
Cocktail tables are very popular among collectors. The main reason is the spouse effect.
You see, a standing arcade cabinet, and a cocktail table both take up about the same amount of floor space. But mentally, the cocktail seems much smaller. This is very important if you have a spouse or roommate who doesn't share your enthusiasm for gaming. They say, "You are not bringing that huge thing into my house.". So you can simply counter with "Well how about one of those small table ones honey?". The mental image of cocktails being smaller (combined with the spouse effect), is the main reason that classic cocktail tables sell for hundreds of dollars more than the exact same game in a stand up cabinet.