ある as a regular verb means “to be”, “to exist”. Basically it still means the same thing when it is used as a helping verb, but it adds an extra dimension.
ある implies that something is in a certain state, because you actively put it in that state.
- 戸は閉まっている -> The door is closed
- 戸は閉めてある -> I have the door closed
- パソコンの電源が入ったままになっている -> The power of the PC is left on (not clear if its is intentional or someone forgot to turn it off)
- パソコンはアップデート中なのでつけたままにしてある -> The PC is left on because it is updating
- 洗濯物は漬け置きしてある -> The laundry is left soaked (for prewash)
- ホテルはもう予約してある -> A hotel is already reserved. I have already reserved a hotel room
- 荷物はもう車に積んである -> The luggage is already loaded in the car
- そのことは彼には言ってある -> I already told him about it. He already knows
- ビールは買ってある -> I’ve already bought beer (and there’s enough on hand)
- 御飯は電子レンジで温めればいいだけにしてある -> The meal is already cooked and you just have to microwave it
ある can be used to mean “to assume, pretend, lie”, like you put the situation in a certain state, if hypothetically.
- 新システム稼働は今のところ四月一日ということにしてある -> We set the startup date of the new system as April 1st for now
- お母さんには全て大丈夫と言ってある -> I told mom that everything was OK (even though it was not really so)
つつある is a literary expression for “be doing”, “have been doing”.
- 地球の平均気温は上昇しつつ有る -> The average temperature of Earth has been rising
The helping verb ある has several other usages, which you can see in the link.
Note ある can be written either 有る or 在る. 有る is usually used to mean both animate and inanimate things to “exist”, while 在る is used for an animate things to physically be at a certain location, even though these are not definitive rules.