A fugue is typical of the Baroque because it revolves around counterpoint (polyphonic textures where different melodic lines form harmonies with one another but can still be identified as distinct 'voices'). Complex polyphonic textures based on a set of 'rules' or principles about the use of harmony and voice leading (the movement of the melody) are very common in Baroque music. The harmony is generally built around relatively simple chords in root position or first inversion, and there is a series of modulations to closely related keys, particularly the dominant. The use of ornamentation in melodies is also a typical Baroque feature.
A gigue is typical of the Baroque in that it's based on a dance form - at that time, a lot of music which wasn't written for the Church was composed either to accompany dances or, more commonly, using the forms of dance music, but not intended for actually dancing to. You can probably identify a lot of typical Baroque features in a gigue (largely diatonic harmony, modulation to related keys, contrapuntal texture, use of binary or rounded binary form, ornamentation etc.). Baroque gigues will usually be part of a suite of dances, often towards the end.
Incidentally, a lot of Bach's gigues actually open with a fugal section, so the two do cross over a bit.