bestellen
to order
To request goods or a service with the intention of buying or receiving them. This often happens in a restaurant, store, or online.
Wir haben zwei Pizzen und eine Cola bestellt.
We ordered two pizzas and a Coke.

to order
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to order
To request goods or a service with the intention of buying or receiving them. This often happens in a restaurant, store, or online.
Wir haben zwei Pizzen und eine Cola bestellt.
We ordered two pizzas and a Coke.

'Bestellen' (to order) is an inseparable verb because it begins with the prefix 'be-'. The prefix is never separated from the verb, and the past participle is formed without 'ge-': 'bestellt'.
Er bestellt sich oft Essen online.
He often orders food online.
The most common meaning of 'bestellen' (to order) is requesting goods or services. Typical combinations are 'Essen bestellen' (to order food), 'ein Getränk bestellen' (to order a drink), 'Waren online bestellen' (to order goods online), or 'ein Taxi bestellen' (to order a taxi).
Lass uns heute Abend eine Pizza bestellen.
Let's order a pizza tonight.
In the phrase 'Grüße bestellen' (to send/pass on greetings), the person to whom the greetings should be passed on is in the dative. The person the greetings come from is introduced with 'von'.
Kannst du bitte deinem Bruder herzliche Grüße von mir bestellen?
Could you please send my warm regards to your brother?
To 'bestellen' (summon) someone to oneself is a formal request and implies a hierarchy. A boss summons an employee to the office, but you don't 'summon' friends to a party.
Der Direktor hat alle Abteilungsleiter zu einer dringenden Sitzung bestellt.
The director summoned all department heads to an urgent meeting.
The idiom 'es ist gut/schlecht bestellt um etwas/jemanden' (something/someone is in good/bad shape) is a fixed construction. It describes the condition or situation of a thing or person.
Nach der langen Krankheit war es schlecht um seine Fitness bestellt.
After the long illness, his fitness was in bad shape.
When someone is officially installed in an office, the structure 'jemanden zu etwas bestellen' (to appoint someone to/as something) is used. The new position or function follows the preposition 'zu' in the dative.
Sie wurde zur neuen Leiterin der Abteilung bestellt.
She was appointed as the new head of the department.
The meaning 'to work/cultivate' is used almost exclusively in agriculture. If you read 'bestellen' together with 'Feld' (field), 'Acker' (field), or 'Garten' (garden), this is the meaning intended.
Der Bauer bestellt sein Feld mit einer neuen Maschine.
The farmer cultivates his field with a new machine.
to buy
'kaufen' and 'bestellen' are both often related to goods, food, or services. The difference lies in the timing and the action: you use 'bestellen' when you request something, for example online or in a restaurant. You use 'kaufen' when you pay for something and then own it. You can order something and buy it later, but an order is not always already a completed purchase. Learners easily confuse the words because in online shopping you often do both.

Im Restaurant bestellt sie eine Suppe und ein Glas Wasser.
In the restaurant, she orders a soup and a glass of water.
Nach der Arbeit kauft die Lehrerin noch ein Heft für die Klasse.
After work, the teacher buys a notebook for the class.