Is it safe to paste a real token here?
Safer than any alternative that involves a text box on the internet, because this page has no server to receive it. The decoding and the verification are done by your browser, and there is no request that could carry a token anywhere — which you can confirm yourself in the network panel rather than take on trust. That said: a token in a browser tab is still a token, so close the tab when you are done, and prefer a test key to a production signing secret wherever you can.
What is the difference between decoding and verifying?
Decoding unpacks base64url and shows you what the token says. Anyone can do it to any token, including one somebody made up thirty seconds ago, and it proves precisely nothing. Verifying recomputes the signature with a key and compares it, which is the step that establishes the token really was issued by whoever holds that key and has not been altered since.
Which algorithms are supported?
HS256, HS384 and HS512 with a shared secret. RS256, RS384, RS512, PS256, PS384, PS512, ES256, ES384 and ES512 with a public key, given as PEM or as a JWK. EdDSA is not supported here yet: browser support for Ed25519 in WebCrypto is still uneven, and a verifier that works in one browser and fails silently in another is worse than one that says it cannot.
My token verified but the tool still says there is a problem.
Then the signature is genuine and something else is wrong — almost always that the token has expired, or that its not-before time has not arrived yet. Those are reported separately from the signature on purpose, because they mean different things: a bad signature means do not trust this token at all, while an expired one means it was genuine and is now past its time.
Do you support decoding encrypted tokens (JWE)?
No. A JWE is encrypted rather than merely signed, needs the decryption key, and has a different structure with five parts rather than three. It is a different tool and it is not this one; a JWE pasted here is recognised and named rather than reported as malformed.