The Daily Show, on Comedy Central, Mon-Thu at 11PM
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Originally hosted by Craig Kilborn (1996-98), who was pretty funny, but sometimes perhaps just a wee bit annoying and, you know, smarmy. But I don't really remember what the show was like, back then, so there's nothing specific I can say about the Kilborn era. (He later became host of CBS's The Late Late Show.)

Jon Stewart (1999-2015)

"The Daily Show" is most identified with Jon Stewart, who hosted it for 16 years. (Some might even forget he wasn't the original host.) It's a satirical news show, which I guess was more about pop culture during Kilborn's era (as I said, I don't really remember). But during Stewart's era, it became more focused on politics and social issues. (There was still some pop culture, at least in regard to some of the guests who appeared on the show. Many of the guests were politicians or in some way related to politics, but there were also entertainers such as actors or musicians.) Anyway, Stewart was terribly funny, but also passionate about many of the stories that were covered, and during his tenure, the show actually had an effect on the world. For many people (such as myself), "The Daily Show" became their main source of news. This is maybe not a great thing, because it is essentially a comedy show. But the sad fact is, the show is often more reliable for telling the truth than most "legitimate" news sources, these days. The show definitely skews liberal, but that doesn't mean Stewart wouldn't call out liberals when they screwed up. If he seemed harder on conservative politicians (and on Fox News), it's because they generally do more asinine, hateful, even harmful things, including outright lying. (Of course, this view leads many average American conservatives to hate Stewart- and if they do, they might as well go ahead and hate me, too, for considering him unbiased. But it is possible to be conservative and still like Stewart, and agree with most of what he says. It's possible to be conservative and still hate a lot of the things conservative politicians do.) I'm not saying Stewart himself was perfect. I didn't always agree with everything he said, or the way he said it. But for the most part I wholeheartedly agreed, and greatly respected him. Stewart retired from the show in August 2015, and I really wanted to see his final episode, but alas, I missed it. (I often missed the show, for various reasons, which I feel bad about, but I watched it often enough for it to be fairly important to me.)

Trevor Noah (2015-2022)
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Trevor Noah began hosting the show in September 2015. And I don't know what to say, except that I think he's very good. The show remains topically very similar to the Jon Stewart years.

Weekly guest hosts (2023)
Following Noah's departure, the show has had a different host each week. I haven't seen all of them, but I'm sure they've all been good. (Certainly the ones I have seen a bit of have all been good.)

Correspondents
Over the years, many "correspondents" and other contributors to the show have become quite well-known, both as an important part of The Daily Show, and for other things. Some of the people I remember best from The Daily Show include:

Lewis Black (1996-present)
Frank DeCaro (1996-2003)
Beth Littleford (1996-2000)
Stephen Colbert (1997-2005; host of The Colbert Report 2005-2014, following The Daily Show on Comedy Central)
Mo Rocca (1998-2003)
Steve Carell (1999-2005)
Nancy Walls (1999-2002)
Rob Corddry (2002-2006)
Ed Helms (2002-2006)
Samantha Bee (2003-2015; host of Full Frontal 2016-present, on TBS)
John Hodgman (2006-2015)
John Oliver (2006-13; hosted The Daily Show for a couple months in 2013; host of Last Week Tonight 2014-present, on HBO)
Rob Riggle (2006-2008)
Larry Wilmore (2006-14; host of The Nightly Show 2015-16, following The Daily Show)
Kristen Schaal (2008-2016)
Jessica Williams (2012-2016)
Jordan Klepper (2014-2017; host of "The Opposition" 2017-2018, following The Daily Show)
Hasan Minhaj (2014-2018)
Desi Lydic (2015-present; host of the special Desi Lydic Abroad in 2019)
Ronny "the Chinese Stallion" Chieng (2015-present)
Roy Wood Jr. (2015-present)
Michelle Wolf (2016-2017)

...And there have been lots of other people whom I don't remember as well (if at all). But they're all good.