When you find something hard to express or difficult to pin down, it's ineffable. The feeling when you get a new puppy is ineffable — too strange and wonderful to define.
You could call something "indescribable", but it wouldn't be quite the same as calling it ineffable. An ineffable feeling, for example, has an almost ghostly quality. You can almost touch it, but it slips away just before you do. The bubbles in a glass of champagne have an ineffable joy to them. The sense of sadness that you feel watching certain TV commercials is often ineffable: you simply can't explain it. You know that strange feeling of satisfaction you feel when you learn a new word? That's an ineffable feeling.