To put everything in simple language: AMD is cheaper, Intel has better performance.
Bay Trail and Clover Trails are code names for chips by Intel aimed at mobile(tablet) market, with Bay Trail more advanced than Clover. They are slower in potential comparing to a Desktop chip but are good with battery life. Ivy Bridge is code name for Desktop/Laptop chips, succeeded by last year's Haswell, which is by leaps and bounds better.
Core (other than the general meaning, which can be thought of as number of brains) is a more specific naming system for Intel. Focus on the Core i3, i5 and i7 series. i3 are mostly found on cheap consumer devices. i5 is a level higher, very capable and powerful with reasonable price; If you are looking for a top notch device beyond average, then there's the i7. Also pay attention to the four digits shown after i3/i5/i7. The 1st of these four digits shows the generation of the chips, with 4 being the newest (ex. i7 4770 is newer than i7 3690).
If you are looking for a tablet device, definitely look for one with Bay Trail.