The resolution of a projector gives you an idea of the image quality it can display. The higher the resolution the better image quality.
It is measured in pixels (a pixel is like a small dot). Resolution is expresed using two numbers: witdh and height. For example, 1920x1080 means 1920 pixels (or dots) in horizontal and 1080 pixels in vertical.
To be practical: If you hire a projector you will need to know what resolution your video or images have. If you want to display a 1920x1080 resolution video (also known as Full HD) you will need a Full HD capable projector. If your image is in low resolution you will not need this kind of projectors and you can get an entry level projector.
When talking about resolutions there are two important concepts we have to look at: Native resolution and Maximum resolution (the maximum resolution a projector can display).
Usually known as SVGA.
This is the resolution a tipical computer used to have. Entry level projectors are designed to work at this resolution. It is an acceptable resolution for Power Point presentations and low quality pictures and videos.
For a non-professional presentation an entry level 800x600 resolution may be enough.
Wide used in digital TV.
Usually known as Full HD, 1080p or 1080i. It is one of the highest resolution nowadays.
You may find more information about resolutions in the Wikipedia.