Maybe. And sometimes No.
The first thing to know is what CPU socket your motherboard has. You cannot put an Intel chip in an AMD socket or vice versa. And within the processor brands, you have to exactly match the processor type to the socket. An Intel LGA1156 chip will not go in an LGA1155 socket for example, nor an AMD FM2 chip in an AM2 socket.
But you are only half done – next you have to check the CPU compatibility charts for the specific model of motherboard. If you have a known brand and model of motherboard, its quite simple, you go to the manufacturer’s website, search for your motherboard model number in the Support or Specifications area, and take the link to CPU compatibility.
If the model number of the CPU that you are considering is on the list, then you can upgrade. Pay attention to the version of the BIOS that you need for the CPU to be supported, sometimes you need to do a Flash BIOS firmware upgrade. This has to be done BEFORE you take out the old CPU. (Always back up your data before doing any operations like updating BIOS or changing hardware).
If the new CPU is not on the compatibility list however, that means that the chipset and the BIOS don’t know how to support the new CPU, and the machine will either fail to boot or will crash. Don’t try to make a non-supported CPU work, its not worth the time or the risk of damaging hardware.
Now, if you have a Dell or HP or Acer or other retail manufactured machine, you have a bigger problem. Not only do these companies usually not publish CPU compatibility lists, but even if they use a motherboard from a company like Asus, Asrock or Foxconn, they have likely custom modified the BIOS. I recommend NOT proceeding, instead put the time and money into a new machine with a current motherboard. If you want to go down this route, you will have to do research online to find people with the exact same machine who have done the upgrade.
Keep in mind that when you upgrade a CPU, you need to re-install your heatsink / cooler. In most cases this means you’ll have to replace the thermal paste with fresh paste. Clean both the top of the CPU and the bottom of the heatsink with alcohol, and put a small amount of thermal paste on the top of the CPU. You can smooth it out to a paper-thin (or thinner) layer using a charge card as a squeegee if you like, and then replace the heatsink carefully, making sure to fasten it down as the manufacturer recommends.
Once you have installed, you may be prompted to go into the BIOS setup to confirm the changes for the new CPU, these could include adjusting the voltage and or the timing. Consult the CPU documentation.