How can you learn organic chemistry nomenclature?

First familiarize yourself with the rules of nomenclature for organic compounds.

Then draw the structural formula of a compound (I'm assuming you have been given a list) on one side of a note card, and the name on the other side, which is a tactile learning style.

Study them, look for the application of the rules to the names of the compounds. Then quiz yourself until you have them memorized. While studying, look for patterns, such as similarities and differences, which involves both a visual learning style and critical thinking.

You can also say the names orally, which will touch on the auditory learning style, and it's much easier to remember a word if you can pronounce it (even if your pronunciation is atypical).

My technique is to go through the cards and set aside the ones that I already know. Then I study the rest and add back the ones I already know. Then I go through all the cards again, and set aside the ones I got right (hopefully there will be more). Then I study the remaining cards, and add back the cards I already know. I keep doing this until I am able to get all of the names correctly from their structural formulas, and understand why they are named that way.

This will take some investment of your time, but it is a very useful studying tool and works for many different subjects.