beseitigen
to remove
To remove or clear away something that is disruptive, unwanted, or harmful.
Die Feuerwehr beseitigte den umgestürzten Baum von der Straße.
The fire department removed the fallen tree from the road.

to remove / eliminate
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to remove
To remove or clear away something that is disruptive, unwanted, or harmful.
Die Feuerwehr beseitigte den umgestürzten Baum von der Straße.
The fire department removed the fallen tree from the road.

'Beseitigen' (to remove/eliminate) is an inseparable verb. The prefix 'be-' always stays connected to the stem 'seitigen', even in a main clause.
Die Stadt beseitigt die Sturmschäden so schnell wie möglich.
The city is removing the storm damage as quickly as possible.
The word can be understood as 'to move aside'. It expresses the idea of clearing away or removing something disruptive.
Bitte beseitige das Chaos in deinem Zimmer, bevor die Gäste kommen.
Please clear up the mess in your room before the guests arrive.
In its concrete meaning, 'beseitigen' (to remove) is often combined with things that are physically disruptive or unwanted. Examples are: 'Müll beseitigen' (to remove trash), 'einen Fleck beseitigen' (to remove a stain), or 'ein Hindernis beseitigen' (to remove an obstacle).
Wie kann ich diesen Rotweinfleck am besten beseitigen?
What is the best way for me to remove this red wine stain?
This meaning refers to non-material things. Typical objects here are 'Zweifel beseitigen' (to eliminate doubts), 'ein Missverständnis beseitigen' (to clear up a misunderstanding), or 'einen Mangel beseitigen' (to remedy a deficiency).
Ich hoffe, dieses Gespräch beseitigt alle Unklarheiten.
I hope this conversation clears up all uncertainties.
In the meaning 'töten' (to kill), 'beseitigen' (to eliminate) is a euphemism, meaning a softened or glossed-over expression. It is mostly used in contexts of crime, espionage, or dictatorships and has a very negative, serious connotation.
Der Agent erhielt den Auftrag, den Verräter zu beseitigen.
The agent received the assignment to eliminate the traitor.
to remove
'entfernen' and 'beseitigen' both mean that something is taken away or removed. 'entfernen' is more neutral and works well when you take something away from a place, for example a stain, a sign, or a file. 'beseitigen' is used more when something is disruptive, dangerous, or represents a problem. It can also fit with abstract things, for example a misunderstanding or a deficiency. If it is only about taking something away from a place, 'entfernen' often sounds more natural. If the goal is to get rid of a problem, 'beseitigen' is more precise.

Julian beseitigt den Fehler in der Aufgabe.
Julian fixes the error in the exercise.
Am Morgen entfernt meine Kollegin die alten Zettel von der Tür.
In the morning, my colleague removes the old notes from the door.