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LingakuLingaku[v5: う-row → え-row] + る [v1: drop る] + られる [する → できる] [来る → 来られる]
The potential form says what someone can do. A う-verb changes its last sound to the え-row and adds る; a る-verb swaps る for られる. する becomes できる and くる来る becomes こられる来られる.
Every potential form is a る-verb, so it changes like one: かけます書けます, かけません書けません, かけました書けました, かけない書けない. The past says what you managed, or didn't manage, to do.
A う-verb never takes られる, and the potential never goes on top of itself.
The potential covers both what you know how to do and what the situation lets you do.
The thing you can or can't do usually takes が instead of を. を also turns up, mostly when the object is long or far from the verb, but が is the safe choice.
With できる, the object takes が.
These go beyond forming and using the polite potential.
In casual speech, a る-verb often drops the ら: たべれる食べれる, みれる見れる, おきれる起きれる, これる来れる. You will hear it all the time, but it still counts as non-standard; in writing and in polite speech use the full られる.
For a る-verb and for 来る, the potential looks the same as the passive: たべられる食べられる is "can eat" or "is eaten". The particles and the rest of the sentence tell you which.