What the finished action left behind
Each one below is our full write-up of that form, in the order worth reading them. Nothing is summarised — you are looking at the same pages the grammar index links to, put next to each other.
do in advance, leave as is
V-て + おく
Do something now because of something later — preparation. 予約しておきます means you book ahead so it's sorted.
It can also mean leaving something alone deliberately: そのままにしておいてください. Casual speech contracts it to 〜とく (やっとく).
has been done (and left that way)
V-て（transitive）+ ある
Something was done deliberately by someone and the result remains. The doer is not mentioned — the point is the present state.
Contrast 〜ている with an intransitive verb: 窓が開いている is "the window is open"; 窓が開けてある is "the window has been opened (by someone, on purpose)".
do completely; do by accident (regret)
V-て + しまう
Two senses that share one form: finishing something off (宿題をしてしまいました), or doing something you regret (財布をなくしてしまいました).
Tone tells them apart. Contracts to 〜ちゃう / 〜じゃう in speech, which is where most learners first meet it.