Important: this page is general information, not legal advice. Copyright law varies by country, and only a lawyer can advise on your situation. With that said, here are the key issues in plain language.
Related: Creative Commons on YouTube — what you can freely convert.
It depends on what you convert and where you live. Converting your own videos, public domain material, or Creative Commons licensed content is generally unproblematic. Downloading copyrighted music you do not own may infringe copyright in many jurisdictions. There is no universal yes or no.
Copyright gives creators exclusive rights over their work, including reproduction and distribution. Downloading a copy of someone's copyrighted song or video without permission can implicate those rights, even if you only listen privately. Enforcement varies widely, but the legal exposure is real in principle.
In the United States, "fair use" allows limited use of copyrighted material for purposes like commentary, criticism, or education. Simply downloading a full song for personal listening is generally not considered fair use. Other countries have their own versions (fair dealing in the UK and elsewhere), which tend to be narrower. "Personal use" alone is not a blanket defense in most jurisdictions.
Separately from copyright law, YouTube's Terms of Service generally prohibit downloading content except where YouTube itself provides a download option (such as Premium offline viewing or an explicit download button). Violating the terms can lead to account consequences. See what YouTube's terms say about downloading.
The original convert2mp3.net shut down in 2019 after a German court found it circumvented technological protection measures, followed by a settlement with IFPI and BVMI. That case was about the service operator, not individual users, but it shows the legal pressure in this space. See what happened to Convert2MP3.
Some material carries little or no legal risk. See what you can legally convert for the practical guide: your own uploads, public domain works, Creative Commons licensed content, and royalty-free libraries.
It depends on the content and your jurisdiction. Your own videos and public domain material are generally fine; copyrighted music you do not own may infringe. This is general information, not legal advice.
YouTube's terms generally prohibit it except where YouTube provides a download option. See YouTube's terms on downloading.
A US doctrine allowing limited use for purposes like commentary or criticism. Downloading a full song for personal listening is generally not considered fair use.
Disclaimer: this page provides general information about legal topics and is not legal advice. Laws differ by jurisdiction and change over time. Consult a qualified lawyer for advice about your situation.