To begin the posts about improvising, we'll describe improvising on what is probably the most commonly-played chord sequence: the twelve-bar blues.
Just in case you didn't know, here's a chord sequence for a relatively basic version of the blues in C (there are in many places in this sequence where different chords could be played):
| C7 | F7 | C7 | C7 |
| F7 | F#dim7* | C7 | C7 |
| G7 | F7 | C7 | C7 | **
*An F minor chord could also be played here.
**In many traditional jazz subgenres, significantly fewer seventh chords are played on a blues.
There are many different ways to improvise over a blues sequence, but here are some general scale-related ideas, or ones that can be used across almost all of the blues. You can use the notes each scale presents in different orders and timing to create practically innumberable phrases:
The minor blues scale. This is probably the most commonly-used scale on a blues, and is good when used to a limited extent. In C, it includes the notes C, Eb, F, F#, G, Bb, and then a C at the top. The minor blues scale can be used over most of the chord sequence shown above. It's best used in a phrase here and there in combination with other, chord-based phrasing.
The major blues scale. This one has a more attractive sound than the minor blues scale, but shouldn't be used just anywhere in the blues sequence by any means. It fits best on the C7 chord during the first four bars; in C the major blues scale includes the notes C, E, G, and A - the same as a C6 chord. You could use the major blues scale's key of F equilavent (F, A, C, and D) on the F7 chord, but still the major blues scale in general shouldn't be used too often.
What is better to avoid when it comes to scales:
Even bebop and hard-bop style musicians should generally avoid using the bebop scale (C, D, E, F, G, A, Bb, B, and C) or the plain C or F scales unless you're used to those scales and can already use them well in improvisation. Otherwise, anyone who is learning to improvise and uses these scales will sound like they are just playing scales, not actual phrases.
Now, here's some improvising ideas from the perspective of chords:
C7 chord: you have a few options here. You can use the notes of the chord (C, E, G, and Bb) for a phrase in basically any order you like, with any timing you want as long as it fits the rhythm. You can combine the notes of the chord with the minor blues scale to get an authentic bluesy sound. You can also use the C6 chord (major blues scale) for ideas; the C6 notes are very similar to the notes of the C7 chord.
F7 chord: you can use the notes of the C minor blues scale or the F7 chord to make a phrase; you can continue a phrase on the F7 chord into the F#dim7 chord or F minor chord to get a fluent-sounding phrase. (Notes of an F7 chord: F, A, C, and Eb.)
F#dim7 chord: this chord includes the notes F#, A, C, and Eb, so it is very similar to the notes of the F7 chord. You can therefore quite easily build phrases from the F7 chord to the F#dim7 chord.
F minor chord: this includes the notes F, Ab, C, and an optional Eb, so it's also very similar to the F7 and once again you can use the similarities to build phrases from the F7 to the next chord.
G7 chord: you'll want to use the notes of the G7 chord (G, B, D, and F) when improvising over this chord.
Of course, many of the improvisation books say and music teachers teach correctly that you have to listen to improvisation to understand how to play phrases in time with the rhythm and know what makes up a phrase.