Improvising is something I cannot imagine playing without. I fall into the category of musicians that virtually `live' on improvised time. :-) I am often surprised that musicians playing straight from manuscript can do so well without being able to improvise effectively. If you have a classical training in music, you may be wondering what it is, how it's done and so on. Of course, there's no answer to this: improvisation is the heart and soul of a good solo, the makings of a musician songwriter and...well, impossible to teach. It involves you taking control of the instrument without music and playing notes based either on the chords, or the melody and timing of the music. In contemporary jazz it's essential to be able to improvise, but you need `good ears'. Listening is the first important thing to master. Singing phrasing and then trying to imitate it can also be an excellent start. The important defining point with improvising is that it can be carried over from one instrument to another, more or less. It's not just a matter of learning riffs on the sax!
If you have a keyboard which will give you a rhythm section, or even play a demo for you, then get that and try playing over what it does. Get a simple 12-bar blues section going, and try just playing the root notes, e.g. E - A - B - A - E as the harmonies progress. Once you've done that, try slipping a few leading notes or trailing notes in, and experiment with different rhythm and tones. Don't try to put down too many notes at once, or do those fast licks like the greats! Take it slowly and see what you can do without technique, just you. Eventually you'll discover what sounds right, and provided you can play in many different keys, you should, with good ears, be able to improvise after a year or so. ;-) I know it doesn't sound like much fun. You can get there much faster, but years will make a solid player while months will give you an excellent head start only.
That's one method. If you've been playing a bit longer, you might think that playing with scales, harmonies and music theory is a little too rigorous and constraining. Some of the best saxophonists in the world think the same, and their works are amazing! You can improvise entirely with your ear and gut feeling. If you've got a good band to play with who can follow your intent closely, then you'll find this immeasurably more rewarding. Such improvisation needs imagination and skill, as well as natural talent and practise.
If you want to try your improvisation out, get a few different copies of a popular, simple song and compare them. Try playing along to them and soloing in different ways. The variation between different interpretations will make you think carefully about what note would suit the version in question, rather than playing blindly.