This idiom or expression has to do with:

Accepting the blame.

When someone says “She took the blame for our failure,” they mean something like “She was blamed, willingly or not, for something she didn’t do.”

Compare to:
Holding the bag.

This idiom or expression has to do with:

Being tricked.

When someone says “I was left holding the bag,” they mean something like “They tricked me and left me to take the blame for something I didn’t do.”

Compare to:
Taking the blame.

This idiom or expression has to do with:

Leaving someone on his (or her) own.
Leaving someone behind to take the blame.
Abandoning someone who is in need of help.

Note:
As this is done by people who know each other, the person who’s left behind feels betrayed.

A. Why are you so upset with me?
B. You left me hanging out to dry!! You didn’t even TRY to help!

Also:
Leaving someone behind.

This idiom or expression means:

A person who finds faults, and blame others, AFTER everything has been done.

A. I would have done the whole thing differently and avoided the present mess.
B. Yeah, it’s easy to be a Monday morning quarterback!!

This idiom or expression has to do with:

Blaming others.
Drawing attention, mainly for blaming purposes.

There’s a lot of finger-pointing going on here, and nobody is willing to take the blame.

Another usage:
Pointing a finger at someone during an argument or while speaking is an insult.

Stop pointing your finger at me! It’s okay if you don’t agree with me, but don’t insult me!