Can is one of the members of modal auxiliary; can, may, must, will/shall. Can usually expresses the idea that something is possible because certain characteristics or conditions exist. Can combines the idea of possibility and ability.

For examples:

Tina is smart. She can complete the test successfully.

The previous statement Tina is smart is to emphasize the auxiliary can complete.... . In this case, there is a possibility of an action which is done by subject, Tina. In other words, It is possible for Tina to complete the test because she is smart. Here, can combines the idea of possibility as well as Tina's ability.

Cannot or Can't
In negation, we can put negative term 'not' in the end of the can. This is to express impossibility of inability.

For examples:

Tina is not smart. She can't complete the test successfully.
or, Tina is stupid. She can't complete the test successfully.

Note: negative can can be written as cannot or can not (separated). cannot is not common in use.

Can Vs May
Can is almost similar to may. may is used to express something possible to do or only 50%  certain to do something.
Compare this follows:

I can be there on time. It's not far from here.
I may be there on time. Or I may go by motorbike.

Able to Vs Can

The use of able to in the simple present (am/is/are + able to) is uncommon (but possible)(Betty SA,106). Be able to is more commonly used in combination with other auxiliaries.

For Examples:

I will be able to finish the job.
instead of, 
I am able to finish the job.