A logarithm can have any positive value as its base, but two log bases are more useful than the others. The base-10, or "common", log is popular for historical reasons, and is usually written as "log(x)". For instance, pH (the measure of a substance's acidity or alkalinity), decibels (the measure of sound intensity), and the Richter scale (the measure of earthquake intensity) all involve base-10 logs. If a log has no base written, you should generally (in algebra classes) assume that the base is 10.
The other important log is the "natural", or base-e, log, denoted as "ln(x)" and usually pronounced as "ell-enn-of-x". (Note: That's "ell-enn", not "one-enn" or "eye-enn"!) Just as the number e arises naturally in math and the sciences, so also does the natural log, which is why you need to be familiar with it.
Warning: If you eventually progress to much-more advanced mathematics, you may find that sometimes "log(x)" means the base-e log or even base-2 log, rather than the common log.
Because the common and natural logs are pretty much the only logs that are used "in real life", these are the only two for which you have calculator keys. Make sure you know where these keys are, and how to use them.
Aside: Why would the "natural log" be denoted by "ln", rather than by "nl"? One popular idea relates to Euler ("OY-lur"), the guy who discovered (invented?) the natural exponential. Euler was Swiss and spoke French, so he might have called the function "le Logarithme Naturel", rather than "the natural log", in which case, "ln" makes sense. However, history shows that Euler actually used just "l(x)" for the logarithm using "his" number e as its base.