Eagle - clearance errors and display errors
If you click on DRC (Design Rule Check) button on the left menu, you can see brief explanations and set the parameters. It is used to see if the factory can manufacture your board(you can ask .dru files from a factor or see their specifications to prepare your own .dru file. Each factor might have different design rules).

Clearance shows the minimum distance between two signals, it is usually 0.2 mm, if your signals are closer than this distance (e.g. 0.2mm), then you need to increase the gap between them.

Dimension shows warnings about the dimensions, As an example, if two component dimensions (red rectangle around the component) are overlapping with another component, then you will have problem on placing the components on the assembly. When you use generic eagle components (resistors, caps, etc) and if you are SURE that dimensions are OK, then you may ignore it, if you are not sure, have a little bit more space between the components (till the red rectangles are not overlapping

Overlap shows that two signals are overlapping, either two traces are overlapping or it overlaps with a pad (you can consider as an unwanted short circuit between two traces). If the paths needs to be parallel to each others and it overlaps, then put them apart, if it is kind of a cross overlap, then route one of the signal from the other layer of the board.

If you click on the error items, it highlights where the problem is. If you are sure that it is not a problem, then you can click on Approve button or you can click on processed.

When there is no DRC error, then you can send it to the factory for the production