Fields are just a special case of rings: for a ring to be a field, the multiplication operation must be commutative and every nonzero element must have a multiplicative inverse. (So, for example, the sets \\(\mathbb{Q}\\), \\(\mathbb{R}\\), and \\(\mathbb{C}\\) of rational numbers, real numbers, and complex numbers all form fields, but the set \\(\mathbb{Z}\\) of integers does not form a field because not every nonzero integer has a multiplicative inverse that is *also* an integer.)

Fields are nice, because (for example) a lot of matrix algorithms require you to be able to divide by nonzero entries, so it's useful to work with fields when you can. On the other hand, those extra axioms mean that you might come across situations where you have a ring but not a field, and then those algorithms don't work.