In general, HTML markup should describe the structure and content of a web page—it should not be used to control presentational aspects such as layout and styling (that's what CSS is for). A tag, like most have already said, should represent tabular data—something that would appear as a table of information.
The reason why people rag on tables so much is that in the old days, there was no such thing as CSS—all page layout was done directly in HTML. Tags were not thought of as describing content—all anyone really cared about was how a tag would make things look in a web browser. As a result of this, people figured that, since they could organize things into rows and columns, tables must be good for laying out elements of a web page. This became a really popular technique—in fact, I'd wager that using tables was considered the preferred method of laying out web pages for quite some time.
So when people tell you that tables are "old-fashioned," they are specifically referring to this abuse of the tag that was so popular back in the old days. Like I said, there's nothing wrong with HTML tables themselves, but using them for web page layout just doesn't make sense nowadays.
(Plus, from a purely pragmatic standpoint, layouts done with HTML tables are very inflexible and hard to maintain.)
