Maybe it's a cliché, but I have always been partial to Piano Sonata #14 in C minor, otherwise known as "Moonlight Sonata" because it is deeply moving to me. As a teacher, I have often played it as part of a CD of background music and kids are always unconsciously humming to it, moving to it, or just stopping to listen.
Many of my own favorite pieces have already been mentioned (especially symphonies 3 and 5), so I'll put in a plug for piano concertos 4 and 5. Let me also recommend symphony 4 (with one of the most clever openings of any symphony I know) and symphony 7 (one of the most rhythmically unforgettable works ever written).
It's no surprise that you get different answers to your question, since Beethoven's compositional output was of the highest quality. I tend to agree with the two previous posts who mention Symphony No. 5 and Symphony No. 9 as his two greatest pieces. I would also add his Symphony No. 3 ("Eroica") as another of his finest works.
I would say probably his Ninth Symphony in D Minor. The final movement, Ode To Joy, is a bit overplayed, but the work as a whole is very significant; it was the first symphony to incorporate a chorus along with the instrumental orchestra, which was very innovative for its time. This link is an interesting article on the piece and its political significance in Beethoven's time.
Well, I am so glad this is in the discussion forum! You are going to receive a wide range of answers, and for me, I would have to say it is his Fifth Symphony. There are many reasons for this, but one of them is the way that it is visually described in one of my favourite books, which is Howard's End by E. M. Forster. Two of the central characters attend a lecture on the kind of images that appear in our minds when we listen to this piece of music, and for me this makes it unforgettable.