Learn German: Ich packe einen Koffer. - I am packing a suitcase.

In this sentence, the suitcase is the direct object (the thing being packed). German uses the accusative case for direct objects. Because Koffer is a masculine noun (der Koffer), its indefinite article ein (a) changes to einen in the accusative case.

The translation says 'I am packing', but why isn't there a word for 'am'?

German does not have a continuous '-ing' tense like English does. The simple present tense Ich packe translates to both 'I pack' and 'I am packing' depending on what makes the most sense in context.

Is there a trick to knowing that Koffer is a masculine noun?

While you generally have to memorize the gender alongside every new noun, there is a handy pattern here: many German nouns that end in -er (especially those referring to tools, devices, or everyday objects like a suitcase) are masculine.

How would I change this sentence to say 'I am not packing a suitcase'?

In German, you often negate a noun by using 'no' instead of 'not a'. To do this, you use the negative article keinen. Just like ein changes to einen for masculine nouns in the accusative case, kein changes to keinen. You would say: Ich packe keinen Koffer.