Central Processing Unit
The processor is the "brain" of a device. It's what handles the instructions of software apps.
All phones have a CPU of some kind. It's typically integrated into one larger chip (called an SoC) that handles many other key functions.
A better and/or faster processor allows apps to run faster. Raw processor speed is measured in MHz or GHz. (1 GHz = 1,000 MHz.) However some processors are more powerful than others, so even if two different processors both run at 1 GHz, if one is more powerful, it may run software apps much faster.
Other factors often affect the perceived speed of a phone. For example, the wireless radio and network determine how quickly data downloads. Also, phones may have a separate graphics processor (GPU) that determines the speed of visual elements, especially with games and other areas with complex animations.