〜まま
as it is, leaving (something) in a state
Verb た-form / ない form / Noun の / Adjective + まま
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まま means "as is", with something left unchanged: テレビをつけたまま寝てしまった, "I fell asleep with the TV on".
It usually follows a た-form, describing the state something was left in: 靴を履いたまま, "with your shoes on"; 窓を開けたまま, "with the window open". Often it's a state that shouldn't have stayed that way.
After nouns it takes の: 昔のまま, "just as it used to be". このまま (like this, as it is) and そのまま (as it is, just like that) are everyday words: このままでいいです, "it's fine as it is".
Compare ながら, which is doing two things at once. まま is one state carrying on while something else happens.
Examples
テレビをつけたまま寝てしまった。
I fell asleep with the TV on.
靴を履いたまま、部屋に入らないでください。
Please don't come into the room with your shoes on.
窓を開けたまま出かけてしまった。
I went out and left the window open.
このままでいいです。
It's fine as it is.
この町は昔のままですね。
This town is just the way it used to be.