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.

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.
last
'letzt' and 'nächst' both show a position in time or in a sequence, but in different directions. 'letzt' means what is at the end or has just passed, for example 'letzte Woche' (last week) or 'der letzte Punkt' (the last point). 'nächst' means what comes after that or follows soon, for example 'nächste Woche' (next week) or 'der nächste Punkt' (the next point). They are easy to confuse because both often appear with time words like 'Woche' (week), 'Monat' (month), or 'Termin' (appointment). Remember: 'letzt' looks back or to the end; 'nächst' looks forward to the following event.

Nächste Woche besucht sie ihre Eltern.
Next week she is visiting her parents.
Beim letzten Termin hat der Arzt nur kurz mit mir gesprochen.
At the last appointment, the doctor only spoke with me briefly.