#1 - The CPU's socket type must match the motherboard's CPU socket type. This means an Intel socket LGA 775 CPU will only fit in a motherboard that is also socket LGA 775.
#2 - Just because the CPU and motherboard have the same type of socket (like LGA 775) does not always mean the CPU and motherboard will always work together. Older LGA 775 motherboards may not recognize newer LGA 775 CPUs. The newerr LGA CPU may physically fit, but if the motherboard's BIOS cannot recognize what CPU is installed the PC will not start.
Example - One of my PCs is built around an MSI 975X Platinum Power Up edition motherboard works with a dual core Intel Core 2 Duo E6600 (It now serves as a home theater PC). However, it does not recognize the Core 2 Quad Q9450 which I used for my next PC. The Q9450 (a 2nd generation Core 2 Quad CPU) can be physically installed in the MSI motherboard, but it's BIOS will not recognize it nor is there an updated BIOS from MSI that would allow the MSI motherboard to recognize the Q9450. Thus, I needed to buy another motherboard that would recognize the Q9450; Abit IP35 Pro. I already knew this and my plan was to have two PCs anyway.