For a laptop, first you need a standard CPU socket to be able to upgrade. If the laptop has one, then you can upgrade to any CPU supported by the BIOS, socket, PSU and cooling system. Note the last one, as most upgrades will generate more heat, and you cannot upgrade your cooling system to be able to handle it. Also keep in mind that laptop CPUs aren't readily available on the market and you have to source them from places like eBay. They aren't cheap.
The process to do it is only a little more difficult than desktops. You have to open up your laptop, usually completely, remove the cooling system, remove the CPU, then put the new one in. Put new thermal paste on it, put the cooling back, close the laptop and hope BIOS accepts it.
In short, it's not worth it.
As for upgrading your desktop, that's much easier. Even if you don't upgrade beyond whatever generation you're in (e.g. Core 2 Duo), you can always upgrade to a higher model (e.g. Core 2 Quad) and just get an after market cooler for that socket (e.g. LGA 775) to handle the improved CPU. Maybe add more compatible RAM (e.g. DDR2) or a better graphics card (e.g. HD 6550). Also make sure your PSU can supply enough juice for the upgrades (e.g. 450W or more).
The easiest upgrade most people can do is buy a SSD and stick Windows on it (and maybe their games, though they can't fit many).