Modal: Can
Modal auxiliary for ability, permission, possibility, requests, and offers.
Can is arguably the most versatile modal in English. It covers three major territories: ability, permission, and possibility — and does so with remarkable economy. One three-letter word carries everything from John can run faster than anyone I know to Can I borrow your pen? to It can get very cold here in winter.
The ability sense of can covers both physical and mental capability. When you say I can swim, you are stating a current skill. Crucially, can is for present and general ability only. When you need to talk about a single past achievement — something you managed to do on one occasion — use was able to or managed to: I was able to fix the sink yesterday. For general past ability, could takes over: When I was younger, I could run for miles.
For permission, can is the everyday, informal choice. Can I leave early? is perfectly fine among friends and colleagues, but in formal situations may is the traditional alternative: May I leave early? sounds more respectful. The distinction is fading in modern English, but knowing both lets you match your tone to the situation.
Can also handles possibility and likelihood. Swimming in winter can be dangerous means it is sometimes dangerous — a general possibility. The negative can't is especially powerful for expressing strong disbelief or negative certainty: You can't be tired — you've been sleeping all day! Here the speaker is certain the listener is not tired. For negative certainty about the past, use couldn't have: He can't have left already; his coat is still here.
Can also thrives in offers and informal requests: Can I help you with that?, Can you pass the salt? It even appears in idiomatic expressions like can do for enthusiastic agreement. Where could is the polite, tentative version and may is the formal version, can is the direct, confident go-to modal for everyday use.
Examples
John can run faster than anyone I know.
Ability
You can't be tired—you've been sleeping all day!
Negative certainty