Bremen is a well-sized city with about 500,000 inhabitants. However, Jacobs University is about 20km away from the downtown area. It is located within a cluster of suburbs called Bremen-Nord. Check out Wikipedia for more info about Bremen!
Jacobs University is located in Vegesack (more specifically in the Grohn part), Universität Bremen in Horn-Lehe, the main train station in Mitte and the airport in Neustadt.
A more detailed map of the Bremen central area can be found on the Bremen city website: http://www.bremen.de/sixcms/detail.php?template=01_stadtteile&gsid=5bremen02.c.927892.de&_subid=927892
Taking into consideration that the Jacobs University campus is in the Grohn area (in the maps above it is somewhere on the edge between Vegesack and Burglesum areas) it might seem a good choice to stay in one of the above two areas if you plan to be on campus very frequently or working late into the night. When looking for apartment close to campus you should consider Vegesack, St. Magnus, Grohn (0-10 min by bus to campus), Aumund (north from Vegesack, appx. 15-20 min by bus from campus), Lesum (south from campus appx. 10-15 mins), Burg (15-20 mins by bus or train from campus), Walle, Findorff (appx. 20 mins by train from campus, depending on the area, you might need to take a bus for 5 mins to get to the train station), Gropelingen (20-30 mins by bus to campus, also good connections to the center of the city, appx. 20-25 mins by tram to the city center).
Disadvantages: very residential area. Far from clubs and bars of downtown Bremen, usually have to return to Vegesack area early in the night.
Many students prefer living in downtown Bremen since it's close to many social and cultural destinations and to Universität Bremen. Mitte (the city center, normally apartments are more expensive here, depending on your location you might need to take a bus/tram to the main train station for 0-5 mins or short walk, and then take the train for 22 mins to campus), Schwachhausen (in the Bremen central area, short walk, or 0-10 min bus/tram ride to the main train station), Horn-Lehe (close to Uni Bremen, 8-15 mins bus/tram ride from main train station), Östliche Vorstadt (5-15 min bus/tram ride from the main train station), Viertel (student area, close to city center, depending on which part of the area the apartment is situated, you might need to take a tram for appx. 5 mins to the city center and then another one for 5-6 mins to the main train station), Neustadt (8-20 min bus/tram ride from the main train station).
Disadvantages: more expensive housing. A 40 minute commute to Jacobs is typical. Also, if you miss the last train from the campus back downtown, usually at 23:27, you have to take the hour long night bus!
Central area: Mitte, Viertel, Schwachhausen, Findorff, Neustadt - from each area it takes 0-15 mins bus/tram ride to Hauptbahnhof (the main train station) and then 22 mins train ride to Schönebeck, the train station close to Jacobs campus.
Around campus area: St. Magnus, Schönebeck, Grohn, Vegesack, Lesum - from each area it takes 0-10 mins bus ride to Jacobs Campus, or a short walk.
Other areas in Bremen-Nord: Aumund - comes after Vegesack, around 15 mins bus ride from campus.
Other areas in Bremen: Burg, Walle, Gropelingen - 10-20 min bus/train ride from campus, normally cheaper than other areas of Bremen, but also more industrial and not very appealing to live in.