gelegen
located
Describes the position or location of something, often in relation to its surroundings.
Unsere Ferienwohnung ist wunderschön an einem See gelegen.
Our vacation apartment is beautifully located by a lake.

located / convenient
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located
Describes the position or location of something, often in relation to its surroundings.
Unsere Ferienwohnung ist wunderschön an einem See gelegen.
Our vacation apartment is beautifully located by a lake.

'Gelegen' is the past participle of the verb 'liegen' (to lie). This origin directly explains the main meaning, which describes the place or position of something.
Das Dorf ist idyllisch in einem Tal gelegen.
The village is idyllically located in a valley.
In the meaning of 'sich befinden' (to be located), 'gelegen' is almost always used after the noun, predicatively. It is often combined with adverbs such as 'zentral' (central), 'ruhig' (quiet), or 'günstig' (convenient).
Das Hotel ist sehr verkehrsgünstig gelegen.
The hotel is very conveniently located for transport.
The expression 'jemandem gelegen kommen' (to suit someone / to come at a convenient time for someone) is fixed. The person for whom something is suitable is always in the dative. The thing that is suitable is the subject of the sentence.
Die Verschiebung des Termins kommt mir sehr gelegen.
The postponement of the appointment suits me very well.
The expression 'jemandem ist an etwas gelegen' (something matters to someone) is very often constructed with 'daran' and a 'dass' clause or an infinitive with 'zu'. It expresses that a specific outcome is important to someone.
Meinen Eltern ist viel daran gelegen, dass ich mein Studium abschließe.
It matters a lot to my parents that I finish my degree.
important
'wichtig' and 'gelegen' can seem similar when something matters to a person. 'wichtig' is the direct and general form: something has great importance. 'Jemandem ist an etwas gelegen' also means that someone places great value on something, but it sounds somewhat more formal. 'Gelegen kommen' is different: it does not simply mean 'to be important', but rather 'to be suitable or convenient'. Learners may confuse this because in both cases a person evaluates something positively. A simple guide: use 'wichtig' for importance, 'gelegen kommen' for a suitable time, and 'an etwas gelegen sein' for personal interest.

Der Ärztin ist an einer ruhigen Vorbereitung des Gesprächs gelegen.
A calm preparation for the conversation matters to the doctor.
Beim Termin ist es ihm wichtig, dass seine Familie gut informiert wird.
At the appointment, it is important to him that his family is well informed.