This picture is a good example of mixing genres. There are loads and loads of shows that mix genres, but usually they stick to mixing two or three to provide a new context or contrast, such as "Brick", which mixes 40s film noir with high school melodrama; "Some Like It Hot", which mixes gangster dramas with slapstick or screwball comedy; "Dancer in the Dark", which mixes Dogme films with musicals.
On my count, however, "Jerry Maguire", to one degree or another, mixes at least six (6!) different genres, necessarily in small degrees.
It is primarily a comedy, and that's due to various jokes and gags throughout, such as most scenes with Rod Tidwell and his outrageousness. But it is also a comedy in the primary, classical sense that it ends happily for all involved. Even the main villain of the film, Bob Sugar, gets off lightly.
That said, there is enough drama that it can't be considered ONLY a comedy. By definition, there are a series of episodes that are turbulent or highly emotional -- Jerry goes through an initial breakthrough (breakdown?), loses his job, his wife, his main client. He can't be alone. He gets involved with his secretary, a single mom. He gets married without being in love. And, most notably, he changes from super-slick and highly cynical to wearing his heart on his sleeve. Sure, it ends well, but he goes through the wringer to get there.
It is also a romantic comedy. You don't shoot a slapstick scene with pasta on the shirt, a declaration of love over morning coffee and dramatic irony, and a three minute neck-kissing scene without being interested in how romantic comedy works. And you certainly don't include "You complete me" without being a complete softie.
So, it's also a chick flick. It concerns the concerns of women, and their problems with men. By definition, it was designed, in part, to appeal to a female target audience. That's why you see Tom Cruise cry so often. That's why you have the voice of Bonnie Hunt, the surrogate voice of many of the women in the audience, so front and center. That's why we see the living room bitch session. And that's why, in the end, love conquers all.
It's also a buddy movie, about the initial animosity and subsequent respect and friendship between Rod Tidwell and Jerry. Rod teaches Jerry about the importance of family and your relationships, and Jerry teaches Rod that there's more to your career than making money at it. And they meet in the middle to become buddies.
And finally, of course, it is a sports film. The third act is almost entirely concerned with The Big Game (Monday Night Football, in this case), a standard sports film trope. And, like almost all sports films, it ends predictably...