Federico Sasso: Both are HTTP status code, returned by a web server upon a client (normally a browser) URL request.
Easier one first: 503 is the way for the server to communicate "Service Unavailable" state. Not properly an error, it's just saying "please return later", likely because the server is temporarily under maintenance. From an SEO perspective, this is important because it's the proper way for a web site to say a search engine not to bother with what it sees at the moment, the real content will be served in a while.
Tough one last: 510 is not part of the original HTTP specification, I never stumbled upon it and had to check what it means. I by no means am authoritative answering this. It basically part of a protocol extension to permit further uses of HTTP; 510 appears to mean that the server didn't receive enough information to fulfill the request with a proper answer.
Ryan Cramer: I'd recommend notifying your host provider about these issues as they are often mis-configurations with the server and/or network routing
Edwin Jonk: From the expert panel in this week's SEO Questions hangout on air on 00:13:30 into the YouTube video: https://dumbseoquestions.com/q/510_and_503_error +arathymol t p
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