My synagogue hired a cantor for the high holy days. (We don't currently have one otherwise.) He's a friendly fellow, obviously very experienced, and very "performative" -- which some people liked but isn't to my taste. (I felt like I was at the theatre.) Unfortunately it's not just a matter of taste; elaborate chazzanut that you can only listen to is fine in a traditional setting, where it's in the cantor's repetition of the central prayer, but the Reform movement did away with repetitions. When there's only one trip through the prayer, everyone saying it together, and it's being led in a way that precludes me saying it, that's a problem. After Rosh Hashana evening and morning were like that, I decided not to go back. (I later skimmed the video of the second-day Rosh Hashana service, which started as a minyan-style service but drifted, and it was more of the same.)

For Yom Kippur I went to Chabad, like I did last year. Night and day -- I felt included from the moment I walked in, I was able to focus on the kavanah, intentions, behind the prayers, the more elaborate melodies didn't impede my own prayer because they were separate from it, a lot of the singing was accessible even with unfamiliar-to-me melodies, and there was plenty of way-finding (page numbers, quick explanations, etc) so people didn't get lost.

All are welcome, all included, on Yom Kippur, the machzor (special prayerbook) says, even transgressors, even that guy. Even me. Read more…

A non-Jewish visitor asked: according to Judaism, can you pray in any language (and have it accepted), or must you pray in Hebrew?

My answer:

The general answer to your question is that Hebrew is preferable but you also need to understand what you're saying, so it's permitted to pray in other languages. (Ideally you are working to improve your Hebrew understanding along the way.)

More specifically: the talmud on Sotah 33a (and vicinity) says that the Amidah, the central prayer of the thrice-daily service, may be said in any language. Many rabbis (who I can't precisely cite, but Maimonides, the Rambam, is one of them) teach that intention in prayer is important; just saying words you don't understand does not fulfill the obligation. Whether you understand the Hebrew directly, have studied a translation and so know broadly what you are saying, or pray in a language you do understand, you need to mean the words you're saying.

Another core recitation -- not technically a prayer but a passage of biblical text, said morning and night -- is the Shema. There is some disagreement in the talmud about whether this biblical text must be recited in the biblical language (Hebrew); I don't know if it's settled there, but the Shulchan Aruch (Code of Jewish Law) in Orach Chayim 62:2 says that the Shema can be said in any language so long as you are precise in your pronunciation.

There are other fixed prayers; I would reason that if other languages are permitted for these "core" cases, then surely they must be for other cases too.

Everything I've said so far is about fixed prayers. Most of (modern) Jewish prayer is fixed texts, but people also pray from the heart, free-form prayer, both during the regular service and at other times. There is a discussion in the talmud (ibid.) about this; some say that the ministering angels (who are apparently involved in bringing your prayer to God) don't understand other languages, and others say that some angels do understand other languages and, anyway, God does. As a practical matter, I don't see how you could pour out your heart to God if you don't know the language, and I've never heard that one should abstain from these personal prayers for lack of language fluency. It is better to reach out to God than not.

As noted in a comment by dsr, what I've said here is the law for Jews. According to Judaism, non-Jews have a much smaller set of commandments, the seven Noachide laws, and they do not include a requirement to pray at all, let alone in a particular language. While there are parts of Jewish observance that are "reserved" for Jews (that is, Jewish law says non-Jews are forbidden to do them), prayer is not one of them. So non-Jews are not required to pray, may pray, and if so may pray in any language they understand, since even Jews can pray in any language they understand.