Learn Polish: Szukam pracy na pół etatu. - I am looking for a part-time job.

The verb szukać (to look for) is a classic example of a Polish verb that requires its object to be in the genitive case rather than the accusative. Even though the job is the direct object of your search, you must say szukam pracy.

What exactly does the word etatu mean in this context?

Etatu is the genitive form of etat, which translates to a full-time employment position or a permanent contract. So, na pół etatu literally translates to "for half of a full-time position."

If na pół etatu is part-time, how would I say I am looking for a full-time job?

You would say Szukam pracy na pełny etat. Notice that pełny etat (full-time) is in the accusative case here, but in our sentence, etatu is in the genitive. This happens because the word pół (half) forces the noun that comes immediately after it into the genitive case.

Could I use a single adjective for "part-time" instead of this phrase?

Not really. Polish doesn't have a common, single adjective that directly translates to "part-time." Describing a job using na (on/for) plus the fraction of the etat is the most natural and standard way to express this in Polish.