By vs. Until

Both words until and by indicate “any time before, but not later than.”

By
By is used in both positive and negative sentences.it is used with verbs referring to one action performed at a specific point of time like:return,pay,finish,end,arrive,...

By in positive sentences

By indicates a deadline and means that an event ,situation or state :

A- does not continue later than a particular time and it happens at or before that time.
- I will return your camera by Friday.
- He left his office by 4 o'clock.
- The train is due to arrive by 9 o'clock.
- We will finish our report by tomorrow.

B- Happen during a period of time.
- We danced by midnight.
- By noon,many people had visited the museum.

By in negative sentences

By in these kinds of sentences is usually used when something is later than than expected.

Until
Until can be used in both positive and negative statements.In positive sentences it means "up to the time of" and in negative contexts it means "at a specified time".
Until is used with verbs showing continuous states or situations like: continue, wait, drive, walk, live,...

Until in positive statements
In these sentences until:
A- emphasizes the amount of time that is left before a certain time:

- We have until 8'oclock to finish the report.
- She has until Friday to deliver her documents.

B- means an event or state that continues to a specified time in the future and then stops:

- We slept until noon.(we slept up to the noon not later)
- They danced until midnight.
- The ticket is valid until March.(it is valid up to March and not latter than it)
- We waited until the rain stopped.
- She didn't come back until the evening.

Until in negative statements

Not until is used to highlight that an event must not happen or become possible before a specified moment in time or before something else has happened but it happens at a particular time or date. As a summary, we can use until in negative statements when something is later than expected,not before a specific time or event.If until is used in negative sentences, it announces a deadline.

- If we don't arrive until five we are late.(if we don't arrive at five OR anytime till five)
- We didn't sleep until noon.(We slept at noon)
- The ticket is not valid until March.(the ticket becomes valid on the first day of March)
- We didn't wait until the rain stopped.(we moved out when it was still raining.)
- She didn't come back until evening.( not anytime before evening but at the evening)
- He won't be back until 7 o'clock.( He will be back at 7 o'clock)

Common errors
Note the following sentences:
- Until the train got into the station, we had been waiting for six hours. (incorrect)
- By the time the train got into the station, we had been waiting for six hours. (correct)

By the time is not compatible with never:
- The baby never slept by the time her mother arrived. (incorrect)

-The baby never slept until her mother arrived . (correct)

References:
Grammar exchange.com
English forums.com