For its negative form, the single word "cannot" is generally used instead of "can not" in formal writing, whereas in speech the contracted form "can't" (/kɑ:nt/) is commoner.
As a full modal verb, it lacks an infinitive, past participle and present participle (see infinitive of a modal verb). However, one can use circumlocution to create an infinitive and present participle; mainly "to be able to" and "being able to" (for ability) but in some circumstances "being/to be allowed to" and "being/to be permitted to" may be appropriate.
The preterite of can is could, and this may be used (1) for past time (2) for irrealis mood. Note that questions in the irrealis mood, e.g. "Could I go home early?" become realis mood when answered and so need to use can - e.g. "No, you can't!".
Native English speaker often subject can to vowel reduction, where /kæn/ is reduced to /kən/. They also often drop the /t/ in "can't" towards a glottal stop /kɑ:nʔ/. Could has vowel reduction to /kəd/ and couldn't not only has vowel reduction but also loses both the /d/ and /t/ sounds to /kənʔ/
Learners may incorrectly reduce or drop the final /n/ sound which is fairly strong, especially before a vowel sound, e.g. "Can I" /kænaɪ/
Can is always the first full modal verb that learners learn, and once they've learnt the basic grammar for can they know the basic grammar for the others, so they may think of other full modals as belonging to the same family.