
Bescheid sagen/geben
to let someone know
To inform someone about something or send them a message.
Kannst du mir bitte Bescheid geben, sobald du zu Hause bist?
Could you please let me know as soon as you get home?
notice / information
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to let someone know
To inform someone about something or send them a message.
Kannst du mir bitte Bescheid geben, sobald du zu Hause bist?
Could you please let me know as soon as you get home?

to be informed
To be informed about a matter; to know something.
Danke für die E-Mail, jetzt weiß ich Bescheid.
Thanks for the email; now I'm informed.

notice
An official, usually written notification or decision from an authority or institution.
Der Steuerberater wartet auf den Bescheid vom Finanzamt.
The tax advisor is waiting for the notice from the tax office.
The most important difference: In the expressions 'Bescheid geben' (to let someone know) or 'Bescheid wissen' (to be informed), the word is used without an article. When it means an official document, it is a normal noun with an article ('der Bescheid', the notice).
Ich weiß Bescheid, danke!
I'm informed, thanks!
The expression 'Bescheid geben' or 'Bescheid sagen' (to let someone know) always needs a person in the dative who receives the information. The structure is: jemandem (dative) Bescheid geben.
Gib mir bitte Bescheid, wenn du fertig bist.
Please let me know when you're finished.
If you want to say what you are informed about, you use the preposition 'über' with the accusative. The structure is: über etwas (accusative) Bescheid wissen.
Weißt du über die neuen Regeln Bescheid?
Are you informed about the new rules?
As an official document, 'der Bescheid' (the notice) is often combined with verbs that describe a reaction. Typical examples are: 'einen Bescheid erhalten' (to receive a notice), 'einen Bescheid prüfen' (to review a notice), or 'Einspruch gegen einen Bescheid einlegen' (to file an objection against a notice).
Sie müssen innerhalb von vier Wochen Einspruch gegen den Bescheid einlegen.
You must file an objection against the notice within four weeks.
In the meaning of 'offizielles Dokument' (official document), 'der Bescheid' belongs to administrative language (official German). You find the word in letters from offices, authorities, or insurance companies.
Laut Bescheid müssen wir eine Gebühr nachzahlen.
According to the notice, we have to pay an additional fee.
answer
Antwort and Bescheid can both refer to a message in response to a request or question. Antwort generally refers to the content of the reply, while Bescheid particularly emphasizes information about a decision or the current status. When you are expecting a response to a request, the fixed expression Bescheid geben (to let someone know) is especially typical.

Auf meine Bitte versprach mir die Mitarbeiterin, bald Bescheid zu geben.
The employee promised to let me know soon in response to my request.
Auf meine Bitte bekam ich am nächsten Tag eine freundliche Antwort.
I received a friendly reply to my request the next day.