The prefix often has a specific meaning, but the numbering following the prefix often doesn't.
1N... = diodes
2N... = transistors
A... (2 letters + 3 digits) = germanium transistor, e.g. AF117
B... (idem) = silicon transistor, e.g BC847
For diodes like 1N400x the last digit is kind of counter to indicate the diodes belong to the same series:
The 1N4148 is a typical switching diode. For it's SMT counterpart manufacturers use the same number (4148), but with a different prefix: Fairchild calls it an LL4148, Rectron an MM4148.
On the other hand, the SMT version of the BC547 transistor is the BC847, so there they keep the prefix, but change the number. You try and find the logic in it.
IC manufacturers often release new devices with their own prefix, like "LT" for Linear Technology, or "LM" for National Semiconductor, so sometimes it refers directly to the name, but often it doesn't. When other manufacturers make compatible parts, however, they often stick to the same part number, so that prefix doesn't always tell you who the manufacturer is.
A MAX809, for instance, is made by (at least) Maxim, On Semiconductor and NXP. "TIP" originally meant "Texas Instruments Power" but you'll also find a TIP110 transistor with Fairchild.
Like Matt says sometimes the number following the prefix refers to the device's function. He mentions the MAX232 as an EIA232 driver, and guess what the MAX485 is. FTDI's FT232R is also an EIA232 bridge. But those are really exceptions.
Sometimes the last digit refers to the number of opamps, for instance, in a device.
LF411 = single opamp
LF412 = dual LF411
I once asked a question about other than manufacturer's prefixes in IC type numbers, but there seems to be little systematical in it.