ARM processors are advanced mostly 32 bit processors, these are basically system-on-chip devices or SOC.
This means the processor is like a tiny computer with quite advanced capabilities. Due to the compact size and capabilities these are used in smartphones and tablets. however due to the popularity and low cost of these chips you can also find these used in medical equipment (like blood pressure monitors, glucometer etc.), fitness equipment etc.
if you are looking into getting into ARM processors, try Cypress (cypress.com), they have cheap development kits that are quite good for entry level and learning purposes. Their PSOC development software (free for commercial use) is quite unique and unlike any other in the market.
The PIC series are micro-controllers, as such are not SOCs like the ARM. However they are quite cheap and are popular for small projects. For more info see Microchip site (microchip.com).
The choice of the processor (whether ARM or PIC) will depend on the complexity, cost and other factors, this is difficult to decide without further information