While we consider these to be barcodes, they do not all use bars and so it is best to use the term symbology to refer to them. There are two types of 2D symbology matrix and stacked (or multi-row) codes.
As with linear symbologies there are many variations of 2D, but only a few of these have been standardized by ISO. The most common ones are:
MaxiCode: a matrix code that is a one inch square code made up from dots (actually hexagons). The symbol contains a bulls eye finder pattern. The primary use of the symbology is by United Parcel Service to encode names and addresses for shipments.
Data Matrix: a matrix code recognized by two of the sides being a solid bar extending the full length of the symbol. Although it is usually square, it does not have to be. The dots that make up the symbol are squares that encode the data in the same way as other symbologies. Fully alpha numeric, Data Matrix is used in many applications and is often seen on pharmaceutical products, part numbering, and many other uses.
PDF 417 is an example of a multi-row symbology. It can be thought of as many linear bar codes placed on top of each other. It is often used on airline boarding passes, drivers license applications, as a postage stamp in the USA, and other places where large amounts of text are encoded.
Other 2D symbologies that are frequently seen include QR Code, Aztec Code, Datastrip, and Grid Matrix.