
nächst
next
Refers to the immediately following point in time or period of time.
Nächste Woche fahre ich in den Urlaub.
Next week I'm going on vacation.
next / nearest
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next
Refers to the immediately following point in time or period of time.
Nächste Woche fahre ich in den Urlaub.
Next week I'm going on vacation.

nearest
Refers to the place or object that is least distant in space.
Die nächste Bushaltestelle ist nur zwei Minuten entfernt.
The nearest bus stop is only two minutes away.

next
Indicates the following element in an order, list, or sequence.
Wer ist der nächste auf der Liste?
Who is next on the list?

closest
Describes the shortest distance to something or a close personal or emotional relationship.
Von allen meinen Freunden steht er mir am nächsten.
Of all my friends, he is closest to me.

the next person
As a noun: the person whose turn comes next in a sequence.
Der Nächste, bitte!
Next, please!

one's neighbor
As a noun in an ethical or religious sense: one's fellow human being.
Man sollte seinen Nächsten lieben wie sich selbst.
One should love one's neighbor as oneself.
The word 'nächst' (next/nearest) is the superlative form of the adjective 'nah' (near). So it always describes what has the shortest distance—whether in space, time, or a relationship.
Wir treffen uns am nächstgelegenen Bahnhof.
We're meeting at the nearest train station.
'Nächst' is very often used as a noun by placing an article before it. 'Der/Die Nächste' means a person, while 'das Nächste' means a thing or action in a sequence.
Was ist das Nächste auf unserer To-do-Liste?
What is next on our to-do list?
'Nächst-' is used very frequently for time expressions. Typical combinations are 'nächste Woche' (next week), 'nächsten Monat' (next month), 'nächstes Jahr' (next year), and 'beim nächsten Mal' (next time).
Nächsten Freitag habe ich einen wichtigen Termin.
Next Friday I have an important appointment.
To describe the nearest place in space, 'nächst-' is often combined with places from everyday life. Common phrases are 'die nächste Haltestelle' (the nearest stop), 'der nächste Supermarkt' (the nearest supermarket), or 'die nächste Apotheke' (the nearest pharmacy).
Entschuldigung, wo ist hier der nächste Geldautomat?
Excuse me, where is the nearest ATM around here?
To express a close personal relationship, the construction 'jemandem am nächsten stehen' (to be closest to someone) is used. The person with whom the relationship exists is in the dative.
Von all meinen Geschwistern steht mir meine Schwester am nächsten.
Of all my siblings, my sister is closest to me.
The call 'Der Nächste, bitte!' (Next, please!) is a very typical prompt in situations with queues, for example at the doctor's office, in a bakery, or at a government-office counter.
Nachdem der Kunde bezahlt hatte, rief der Kassierer: „Der Nächste, bitte!“
After the customer had paid, the cashier called, 'Next, please!'
The meaning 'Mitmensch' (fellow human being) is used mainly in an ethical, philosophical, or religious context. The related concept is 'Nächstenliebe' (love of one's neighbor/charity).
In seiner Rede betonte er die Wichtigkeit, sich um seinen Nächsten zu kümmern.
In his speech, he emphasized the importance of caring for one's neighbor.
following
Nächst and folgend can both refer to the immediately following period of time. “Nächste Woche” is usually understood from the current time of speaking. “In der folgenden Woche,” by contrast, typically refers to a point in time mentioned earlier and is particularly common in reports.

Nächste Woche beginnt der neue Kurs.
The new course starts next week.
Sie meldete sich am 3. Mai an, und in der folgenden Woche begann der neue Kurs.
She registered on May 3, and the new course started the following week.