The definition is vague, so who is a grappler may be recognized differently by different players, but it is generally agreed that there needs to be at least two kinds of command throws (i.e., at special move level or higher). Nearly all grapplers will have at least one throw for a super special move.
The first character recognized as a grappler is Zangief from Street Fighter II (and he conforms to this classification). In most cases, grapplers will be thick-boned and/or muscular men. This stereotype justifies their high offensive output and defensive ability, as well as their low speed, jump ability, and and offensive throughput, around which their play strategies must center. Naturally, more recent games have made stark contrasts to these characteristics (best [counter-]example? Kira, a swimsuit-clad 11-year-old. Not my cup of tea in the least.).
Also, they grapplers are generally bad at far distance; it is necessary to approach in order to get near enough to throw, so it is important to check the opponent’s actions with zoning and such. The easier-to-use grapplers tend to be so because their normal moves have good properties, allowing one to aim for throws using those alone.
If one does not or cannot approach, the true value of grapplers are not immediately apparent, so they are essentially for advanced players in a sense, but some games do make things easier on beginners (e.g. The King of Fighters has quick jumps, allows throw attempts from an emergency evasion, and allows one to grab an opponent’s emergency evasion).
Because there is a healthy number of grapplers, and because new games come out every once in a while, it is my regret to inform you that not all grappling characters neither can nor will be be listed here.