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LingakuLingaku[v1: drop る] + させる [v5: う-row → あ-row] + せる [する → させる] [来る → 来させる]
The causative says that someone makes or lets another person do something. A る-verb drops る and adds させる; a う-verb moves its last sound to the あ-row and adds せる – a verb ending in う takes わ, so かう買う becomes かわせる買わせる. する becomes させる and くる来る becomes きさせる来させる (こさせる).
The causative verb is a る-verb and conjugates like one: たべさせます食べさせます, たべさせた食べさせた, たべさせない食べさせない.
The one who makes it happen is the subject. The one who does the action takes に when the verb already has an object with を, since a sentence can't carry two を.
When the verb has no object, the person can take を or に. を leans towards making them do it; に towards letting them, or getting them to do it willingly.
The causative alone does not say whether the person wanted to. Context decides, and so does what follows. With てくれる or てあげる, it is clearly "let", a favour.
These go beyond making or letting someone do something.
With a verb of feeling, the causative means "cause": you make someone wait, worry, laugh or cry, often without meaning to.
The causative て-form with ください asks for permission to do something yourself. It has its own entry, させてください.