Brooklyn is covered by an extensive network of MTA buses, for which a map is essential. http://mta.info/nyct/maps/
Of particular note is the B51 bus, which runs between City Hall in Manhattan and Smith St./Fulton St. in Downtown Brooklyn, via the Manhattan Bridge. The trip is particularly beautiful on the way to Manhattan. Note that the bus operates only on weekdays, with the last bus leaving Smith St./Fulton St. at 7:10 P.M. and from Park Row at 7:40 P.M., according to the current schedule and depending on traffic. See the MTA website for bus maps and schedules of individual bus lines. The B39 travels over the East River on the Williamsburg Bridge between Manhattan and Brooklyn.
Another route of note is the B15, which runs between JFK Airport and the Woodhull Hospital in South Williamsburg. Service on that route is provided around the clock. Possibly the longest mostly straight-line bus route in Brooklyn is the B41, which runs nearly the length of Flatbush Avenue from the line's northern terminal near Borough Hall just west of Flatbush Avenue itself to Kings Plaza at Avenue U (with a second branch to the Mill Basin area), about 9 miles away.
Other long routes run a Limited-Stop service making stops at major intersections and points of interest. Limited-Stop service is provided on the B6, B41, B44 (along Nostrand/Bedford Avenues), B46 (along Utica Avenue), and B35 (along Church Avenue). That pattern is in effect from around 6.00am to 10.00pm daily on these routes (5.00am-11.30pm on the B46). The B49 has southbound limited-stop service on weekday mornings, mainly tailored for college students traveling to Kingsborough Community College. The B103 is a limited-stop route from downtown Brooklyn to Canarsie, which runs on weekdays and Saturdays.
One can also take express buses to/from Manhattan ($5 one way). Most express buses serve Southern Brooklyn, for the most part an area that's somewhat subway deprived, especially to the east. The X27 and X28 run daily from around 6.00am to 11.30pm. The X29 runs rush hours only, to Manhattan in the morning and to Brooklyn in the afternoon. Other express routes run every day except Sunday, and have a BM prefix.