zurückbleiben
to stay behind
To remain in a place while others leave.
Als alle Gäste gegangen waren, blieb er allein zurück, um aufzuräumen.
When all the guests had left, he stayed behind alone to clean up.

to stay behind / fall behind
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to stay behind
To remain in a place while others leave.
Als alle Gäste gegangen waren, blieb er allein zurück, um aufzuräumen.
When all the guests had left, he stayed behind alone to clean up.

'Zurückbleiben' (to stay behind / fall behind) is a separable verb. The prefix 'zurück-' goes to the end in a main clause. In subordinate clauses and in the infinitive with 'zu,' the verb stays together.
Warum bleibst du immer zurück?
Why do you always fall behind?
The perfect tense of 'zurückbleiben' (to stay behind / fall behind) is always formed with the auxiliary verb 'sein' (to be), because it describes a change of place or state. So the correct form is 'ist zurückgeblieben' (has stayed behind / has fallen behind / has remained).
Nach dem Fest ist ein großes Chaos zurückgeblieben.
After the party, a huge mess remained.
In the figurative meaning 'to drop in performance,' 'zurückbleiben' (to fall behind) is often used with the preposition 'hinter' (behind) and the dative. You fall behind a person, a group, or an expectation.
Das Unternehmen ist technologisch hinter der Konkurrenz zurückgeblieben.
The company has fallen behind the competition technologically.
The command 'Bitte zurückbleiben!' (Please stand back!) in the imperative is a standard announcement at German train stations. It asks passengers to keep their distance from the platform edge when a train is arriving.
Am Bahnsteig ertönte die laute Durchsage: „Vorsicht am Gleis, bitte zurückbleiben!“
On the platform, the loud announcement sounded: 'Caution at the track, please stand back!'
to stay / remain
'bleiben' and 'zurückbleiben' can both mean that someone or something stays in a place. 'Bleiben' is general: a person does not leave, or something is still present. 'Zurückbleiben' is more specific. It often says that others leave and a person or thing stays in that place. It can also mean that something remains as a leftover. The words can be confused because both have to do with not going away. If you are simply saying that someone stays in a place, use 'bleiben'. If the contrast with other people or things is important, 'zurückbleiben' fits better.

Paula bleibt am Bahnhof zurück, während die anderen schon in den Zug steigen.
Paula stays behind at the station while the others are already getting on the train.
Die Ärztin bleibt im Büro, weil noch ein Patient wartet.
The doctor stays in the office because a patient is still waiting.