It depends on the context, but in the RF context an (RF) signal generator is usually a generator that will generate narrow bandwidth signals with a high stability over a fairly wide frequency range, starting around 1 MHz and going up to GHz in some cases. Output types are usually CW and various types of modulation. The output levels might go from +20 dBm down to well under -100 dBm (i.e. a wide range).
A function generator is a jack of all trades. Frequency range is lower than an RF signal generator, from < 1 Hz up to a few MHz up to a few dozens of MHz (though you can get them up to higher frequencies). Stability (e.g. phase noise) will often be worse than a true RF signal generator. Signals can be at least sine (CW), triangle and square wave. Some have more like ramps and pulse trains, and some can even do arbitrary signals within their bandwidth / memory depth. Modulation, if available, is more limited than an RF signal generator. Power output might go from +20 dBm down to -20 dBm, a much smaller range.
A pulse generator is specialized, like an RF signal generator, but more aimed at digital circuits. It can generate pulses with variable transition times (can go much faster than a function generator) and a wide range of duty cycles (wider than some function generators). The signal will likely be more stable (less jitter) than a function generator. Some can even generate variable pulse patterns to emulate digital signals.
So in short, a function generator can do it all but less well and over a more limited range, and the other two instruments are both more specialized and do one particular job very well.