Questions in the title. What is the significance of this term and how it is used by the Buddha in the suttas?
Together this would make Tathagata mean "Firmly established in reality".
I personally associate this word with two other terms:
From an academic perspective the wikipedia article "Tathāgata" has good information. According to wikipedia:
The word's original significance is not known and there has been speculation about it since at least the time of Buddhaghosa, who gives eight interpretations of the word, each with different etymological support, in his commentary on the Digha Nikaya, the Sumangalailasini:
Buddha uses the term to refer to himself rather than saying, I, me or myself.
1. Why does the Buddha call him self Tathāgata?
What would be better than showing the man's own words?
In addition to the above, DN 29 provides one more definition:
2. What is the significance of the term?
As the Wikipedia states:
So it appears these two meanings can be derived from the word. I think some light can be shed by seeing the word used in another context in which there can only be one meaning due to the nature of it's context (simile):
Note that footnote nº2 of the above discourse is also worth reading to get a better grip on it's meaning.
3. How is the term used by the Buddha in the suttas?
As user70 pointed out, the Buddha usually uses the term to refer to himself. Above, I quoted a number of prosaic contexts that reveal how he uses the term, so I'll leave a poetic reference for this answer.
A Tathagata is one who is neither coming or going. That is neither going to the other side or coming back. Because there are no longer sides. Just what is.
Unrul3r's answer includes that it might mean going or coming (gata or agata).
Andrei's answer is that it means 'grounded' or 'rooted' (which makes sense to me: because if you go or come, have gone or have come, to somewhere then that is where you are).
Combined (and further to user319's answer) these remind me of a Zen story -- using Google I cannot find a reference to it at the moment, so I paraphrase it:
tatha: (being) in truth，truthful； true，real
gata: directed to
protected by Andrei Volkov♦ Oct 21 at 17:18
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