There are several types of standard pieces, characterized and identified by the number of studs that piece contains. The "studs" are the little bumps on the top of almost every element in the LEGO system. A stud fits into grooves found on the bottom of almost every other element in the LEGO system. Obviously, a stud can only fit in a groove and vice versa, and so is the most basic understanding in building with LEGO.
This means that as such LEGO models can only be built "one way up", but there are also special parts to work around this and allow for a variety of angular designs.
LEGO bricks are referred to by mathematical proportions much like timber; for example a "2x4 brick" is a brick that is 2 studs wide and four studs long. The 2x4 is the first and most common LEGO element of all. For most applications you will end up needing greater quantities of this brick than of any of the other elements.
There are two basic types of LEGO elements, "bricks" and "plates". Put simply, bricks are the thick elements and plates are the thin ones; plates are always 1/3 of the height of a brick, so a mixture of plates and bricks can be used, but they still meet at the same height.
In special cases a plate-based part may be 2/3 of a brick, or a "brick" may actually be several bricks high itself, and these can allow for greater flexibility in construction.
LEGO pieces come in a wide variety of colors. Basic bricks and plates are available in around 64 colors, while some specialized elements only come in a few colors. Specialty chromed and glow-in-the-dark elements are also available, as well as transparent varieties of many of the solid colors.