For some reason, Bach is better appreciated by learned musicians. Talking to any well-trained musician who plays a lot of Bach, you will realize that it is fully possible to make an emotional connection with the music. In fact, I would argue that Bach alone has written the very most inspirational music, ever.
Here's why: Bach was a genius. As user Widor (impostor!) suggested, the untrained ear can find a little beauty in J.S. Bach's music, perhaps, if they are listening to the overall sound. But listening to each individual note is where the beauty actually happens. (By the way, I think there is a technical aspect to this question and that it is still on-topic). Bach wrote contrapuntal music, which is known to be very organized and "strict". The great thing about Bach is that he knew all of the rules, but broke and bent them in just the right ways to create something altogether new and unfathomably musical and not very strict at all. So you have to listen to each voice and layer of the music and hear their distinctness. Every single note and phrase has a special musical significance.
The sheer variety is endless, and the creativity and development of the music is mind blowing. Often, you hear snippets that really sound like Beethoven or Mozart, or even Chopin; things we don't see again for at least another hundred years: that's Bach being inventive. Many musicians and historians think he was God-inspired. Indeed, there are many theological aspects of Bach's music, just look at the Passions or the Orgelbüchlein.
Across the board, in the other answers and comments, you will hear the same thing: "You gotta listen to this stuff! Here's a list!". I'm going to repeat it. You won't learn how to listen to Bach unless you practice listening to it, just like you won't learn how to play a piece without practicing it. You can start anywhere, because while some Bach music is more complex than other Bach music, it is all equally beautiful, and you are trying to learn to appreciate the beauty here. I had the same question as you about Bach when I was a thirteen-year-old organist and my teacher gave me a Bach CD: After that I was scared I might have been actually addicted to the stuff. ;)