Same as above: quoting something in a language I don’t understand. The symbols are there, and something (the original idea) was communicated–but it wasn’t my thought.
As for ideas that I can’t communicate… well, frankly, just about any kind of feeling that isn’t easily relatable to something more common. The very best poets just barely manage to transmit a scrap of an approximation of a feeling, which really is impressive enough. I’m not among the best poets of all time, and so I’m incapable of communicating such feelings in anything but the crudest possible manner.
I couldn’t disagree more. Communication is about transmitting information through a lossy channel (i.e., our senses). Thought is about information processing. It can take thought to condense and organize information for the best chance at successful communication, of course. And to some extent, we think in symbols, but that is by no means universal.
If I am thinking about “the indefinite integral of x squared with respect to x”, I am not generally thinking about those words, unless I am preparing to talk to someone about it. I am probably not even thinking of \int{x^2}dx, although I can do that as well. Instead, it brings to mind the idea of an integral, which can lead to notions of area and volume, shapes composed of slices of lower-dimensional things, etc. And the idea of x^2 brings to mind actual squares, how they grow and shrink with side length, and graphs with an upward curve, and so on.
In this case, I’m of course capable of distilling it down into something I can communicate, but that isn’t how I’m thinking about it. Really it’s no different from any other experience you might have, whether it’s your response to a smell or sound or something abstract like nostalgia. Even if you can describe the experience, you aren’t really communicating how you think about those things. The best you can hope for is to produce a complete enough sketch that the other person can compare their own experiences and get some inkling of it (or bring to mind the actual experience they had).
Really, it’s amazing that we can communicate at all. Consider the difference in bandwidth: speech is only a few dozen bytes per second, whereas the brain has billions of neurons, and trillions of interconnections, firing dozens of times per second. It must be terabytes per second of activity. We can only communicate the most infinitesimal fraction of what we think.