Learn Danish: Skær løget og kartoflen! - Cut the onion and the potato!

Skær (cut!) is the imperative or command form of the verb at skære (to cut). In Danish, you make a command by dropping the final -e from the infinitive verb, which leaves you with the short bare stem.

Could I say skære løget instead?

Not if you are giving a direct command! Skære is the infinitive (to cut). If you want to tell someone to do it, you must use the imperative form skær. You would only use skære in sentences like 'I want to cut the onion' (Jeg vil gerne skære løget).

Why do løget and kartoflen have different endings if they both mean 'the'?

They belong to different noun genders. Løg is a neuter noun (et løg), so it takes the -et ending to mean 'the onion'. Kartoffel is a common gender noun (en kartoffel), so it takes the -en ending to mean 'the potato'.

Why is 'the potato' spelled kartoflen instead of kartoffelen?

When a Danish noun ends in -el, like en kartoffel, it usually drops the last -e- when adding the -en ending to make the word flow better. Notice that one of the fs is also dropped because the syllables have shifted!