〜てある

has been done (on purpose)

Describes a state that someone created on purpose and left that way: "has been done", "is (done) and ready".

Readings

How to form it

Put a verb that takes an object in the て-form and add ある: 開(あ)ける becomes 開(あ)けてある.
The thing that was acted on usually takes が: 窓(まど)が開(あ)けてあります (the window has been opened).
Who did it does not matter and is not mentioned.

窓(まど)が開(あ)けてあります。

The window has been left open.

テーブルに花(はな)が飾(かざ)ってある。

Flowers have been put out on the table.

ておく is about doing something in advance; てある is about the state it leaves: ホテルを予約(よやく)しておいた (I booked the hotel in advance) → ホテルは予約(よやく)してある (the hotel is booked).

Using てある with a verb that has no object. Say 窓(まど)が開(あ)いている (the window is open), not 開(あ)いてある.

〜ているis doing / is in a state
With a verb that has no object, a state without saying anyone caused it: 窓(まど)が開(あ)いている.
〜ておくdo in advance
Doing something in advance to prepare.