The verbs “nehmen”, “holen” and “abholen” can be very close in their meaning. Depending on the context there are several usages for each.
Here are explanations for each verb and common usages:
Nehmen refers to picking something up that is physically within your reach. It can also apply in other contexts such as ordering in a restaurant.
Picking something up from a table/shelf/…
Ich nehme das Brot. – I take the bread.
Ich nehme die Gabel. – I take the fork.
Ordering a dish in a restaurant
Ich nehme das Schnitzel. – I take a Schnitzel (ordering a dish).
Ich nehme das Erdbeereis. – I take the strawberry ice.
Holen also means that you’re going to pick up/take an item. It implies however that you will need to move physically to get it.
So it can involve walking to another room or at the very minimum getting up from your chair and walking a few steps.
Getting something from another place
Kannst du das Handy holen? – Can you get the mobile phone?
Wir holen den Brief aus dem Briefkasten. – We are getting the letter from the mailbox.
Hol bitte die Milch. – Please get the milk.
A quite similar verb is “abholen” – it refers to picking up someone or something up. Usually it will be there waiting for you and you are supposed to bring the item or person to another place.
Picking someone/something up from another place
Kannst du mich vom Flughafen abholen? – Can you pick me up from the airport?
Ich hole dich ab. – I pick you up.
Das Auto ist repariert, ich hole es ab – The car is fixed, I pick it up.