Pixels

The pixel art refers to a technique used since the early stages of videogame making, and consists of drawing characters, game objects, and backgrounds by drawing on every single pixel of the final bitmap.

Since the computational power to manage graphics was reduced and the drawing tools were not very sophisticated at that time, this was an optimal choice to produce nice graphics in small sized files. The main drawback with this technique is that bitmap images cannot be scaled without losing the details and quality, due to a phenomenon called anti-aliasing. You can find more information on this on Wikipedia at: http://en.wikipedia.org/wiki/Aliasing

The following is a screenshot of a popular mobile game called Sword and Sorcery made with pixel art technique:

Vectors

As the computational power of gaming devices improved and better drawing tools became available, artists turned to a different technique called vector graphics. In vector graphics, every line drawn by the artist is transformed into a mathematical function. As such, vector graphics can be scaled at will without any loss of details. Software, such as Illustrator and Flash, work with vector graphics to suit the needs of developing digital artworks for the Web and print use and almost all web-based games make use of vector graphics to allow full scalability of the graphic assets. On the other hand, vector graphics files tend to be larger than their bitmap counterparts.

The advent of mobile games gave new life to the earlier technique of pixel art. The reduced dimensions of mobile phone screens and the smaller size of graphic files made with this technique turned pixel art into a useful tool to produce nice looking graphics of smaller file size, when compared to the more advanced vector graphics.