Historical city center: (San Miguel, La Catedral, Plaza Circular, etc) Typically more expensive and older than the outskirts, but better connected. Tons of grocery stores, bars, clubs, monuments, etc. Apartments may or may not be a tad smaller. Less parking if you have a car. Much louder if you don’t sleep well with outside noise. If you like the city life, this is for you!
La Flota: Bigger spaces, typically cheaper. Quiet on the streets, but in most buildings you can hear too much of your neighbors. A 15 minute walk from the city center, however, that walk makes a big difference between teaching your private classes or coming home from a night out. A bit farther from grocery stores (especially the fruit/veg stores). Family oriented.
Barrio del Carmen: The neighborhood right below the river. A 10-15 minute walk from the center. Cheaper, usually older apartments. Still well-connected to the center and to shops, grocery stores, fruterias. Closer to the train station zone.
San Andres: Near the bus station, west of the city center. It can get a bit sketchy closer to the bus station at night. Neighborhood is less well-kept, but fine and central.
San Anton/Near “La Opinion”: A small step up (both figuratively and on a map) from San Andres. Various parks in the area. Well connected.
La Fama: The only neighborhood Spanish people have warned me not to live. High drug-use in the area. Not recommended. I personally live on the edge of La Fama, near Plaza de Toros, which I really enjoy. However, I do notice the change when I walk a few blocks towards La Fama.
There are many other neighborhoods, but you can get the general gist of the area from some of the descriptions above. I recommend arriving before your appointment time (to view an apartment) and taking a walk around to form your own opinion of the area as well.