Learn German: Ich suche einen gebrauchten Tisch für den Balkon. - I am looking for a used table for the balcony.

In this sentence, Tisch (table) is the direct object, which puts it in the accusative case. Because Tisch is a masculine noun (der Tisch) following the indefinite article einen, the adjective describing it must take the -en ending.

The English sentence says "looking for", but the German sentence doesn't have a word for "for" after suche. Why?

The German verb suchen translates directly to "to look for" or "to seek". You don't need an extra preposition. You just put the thing you are looking for right after the verb as a direct object in the accusative case (here, einen gebrauchten Tisch).

Is gebrauchten related to a verb? It looks like a past tense word.

Exactly! It comes from the past participle of the verb gebrauchen (to use). Just like in English where you can say "a used table", German often turns past participles into adjectives. The base adjective is gebraucht, and the -en ending was added to match the case and gender of the noun.

Why is it für den Balkon and not für der Balkon?

The preposition für (for) is a strict "accusative-only" preposition in German. This means whatever follows it must always be in the accusative case. Since Balkon is a masculine noun (der Balkon), its article changes to den.