A tonal center (aka a key center, a tonic center, etc) is a groups of chords in a piece that give a strong impression of being in a certain key regardless of what key the song is actually in.

There are used quite a lot in jazz and blues but can be used for any type of music except classical which is slavishly devoted playing the note of the written page exactly as it is written. Key centers are useful for improvisation and most forms of modern music resort to some amount of improv. Key centers often do not even have notes on the staff. They aren't necessary. Really, the time signature isn't necessary most of the time since, in modern music, were are usually dealing with 4/4 and occasionally 2/4 and 3/4. A more exotic time would require the time to be written down. Usually, when you see a chart, the key signature and time are included for reference but aren't necessary.

When I play off a chart, I don't even look at the key signature since it doesn't matter. If you're playing an old folk standard, you don't want some sheet music. You want to know what chords you're going to be playing but you want to put them together your own way. Who wants to play it exactly the same way Dave Van Ronk or Pete Seeger did it? Maybe you liked their version and decided to do that song but it's such a standard that you want to put your own stamp on it. Tonal centers will allow that.

What you have to know before you start is your scales and particularly your chords. You have to know those by heart. And you have to know structure. I was taught it by rote but I found that the circle of fifths helps. Here's a key center chart:

http://www.guitar4free.com/images/jazz-blues-in-f.png

This one is a single tonal center and it's been mapped out for us. The first bar is F7, the second is Bb7. Notice F as tonic going up to Bb is a perfect 4th (picture the circle of 5ths and notice that F and Bb are next to each other counterclockwise which is the "fourths" direction). Now it just alternates between IV and I for a bit and then goes to a VI before shifting to a G-7 chord. Then the G-7 jumps to a C7. Again, G and C are next to each other on the circle of 5ths. with G as the tonic in this interval it is a fourth. G is the ii and the C is the V. But of what? Of the key center--which is what?

Well, if G is ii in the scale then what is I? F. So the key center is F. Then notice the progression of the last line: ii-V-I-VI-ii-V. In modern music, a huge number of songs are ii-V(7) but many are ii-V(7)-I. The VI (which is usually minor but is dominant 7 in this case because jazz likes to do that) is thrown in and many songs are also I-vi-ii-V (in fact, this is called the "doo-wop progression" because that's how doo-wop is structured--I-vi-ii-V).

Notice how it doesn't matter a wit what the key of the song is. All you need to look at is those chords. All the accidentals and what not will take care of themselves. When you want an accidental, you do what this chart does in the second bar of the second line and write a a diminished or augmented chord.

The beauty of the key center approach is that you break the song down into it's harmonic structure which always follows along the circle of 5ths. You can embellish the chart anyway you want and you can do it on the fly which is why it is so useful for improv.

Notice it doesn't even have notes. Why? Because we don't care. We don't need to care. The chords tell us what notes are available for each bar and we can put them together as we please