Here is a list of my favorites, and while lengthly obviously doesn't include everything. "Best" is a very subjective term and so if you want the objectively best I would check out progarchives variety of best-of lists.
In order of release:
Some would argue it's not prog, but this album helped drive a resurgence of experimental music in the UK and US, and a lot of prog bands began popping up again in the 2000s as a result)
Magma's comeback album, taking the helm again as the leaders of the strange prog-derived genre of Zeuhl, and probably one of their best albums.
Battles, though not strictly prog, brought back a derivative genre, Math Rock, back to the popular eye with their complex and innovative music.
Brighteye Brison - Believers & Decievers
This is the first band on this that I would classify as prog and the release around this time was part of a big resurgence. Many prog bands started showing up around this time that were more heavily influenced by classic prog than progressive metal of the early 90s/late 80s.
I've met two kinds of people: people who have never heard of Moon Safari or heard their music, and people who absolutely love this album. Now while it isn't the proggiest of albums, I have never heard as much ear candy for prog lovers on an album before this one, and none since. Unless you really just don't like positive sounding music, it's really hard not to fall in love with this album, and probably one of my favorites on this list. To this day it is my favorite prog double album.
While the resurgence of prog in the late 2000s generated a bunch of great bands, most of which are still releasing albums today, it also generated a bunch of short lived bands that released one really awesome album and then vanished. Moth Vellum was an amazing one-off band like this. Imagine if Yes and Rush had a baby, and you would get Moth Vellum.
Phideaux is every prog band you have ever heard from the 70s rolled into one. This band had already been putting out prog albums but Number 7 was my first favorite of theirs.
Motorpsycho - Heavy Metal Fruit
Motorpsycho made a huge comeback with this album. They were already no stranger to heavy prog, but this album showed they were a force to be reckoned with again. Pretty much every year since this release they have been releasing awesome heavy prog albums.
Neograss - The Sea of Tranquility
This band is an attempt at a new genre of music that combines elements of prog with elements from the bluegrass movement. While they never really took off, their albums include an interesting combination of both prog sounds and instruments with that of bluegrass.
Brighteye Brison - The Magician Chronicles Part 1
Another amazing album from this band. Still waiting on part 2! This is another one of those bands that likes to combine elements from all of the bands of the 70s into one, and you'll hear influences from King Crimson, to Yes, to Gentle Giant, to VDGG, and more, all rolled into one. Their albums all tell a story, this one in particular, all chronicling the saga of their titular character, Brighteye.
Fleet Foxes - Helplessness Blues
The indie rock scene helped drive a rebirth of prog-folk, and this was one of the most creative and beautiful sounding albums to come out of that particular side of the scene. While they take inspiration from bands like Strawbs and Crosby Nash and Stills, they are really their own element. It will be a while before they put out another album unfortunately, if they ever do, and are on hiatus currently.
A lot of prog metal bands around this time began moving away from some of the traits of progressive metal that had turned into gimmicks. This is the album where Haken ditched a lot of those gimmicks in favor of innovation and getting back to what made prog rock what it was, while still being an prog metal band. This album was hugely popular among prog and metal enthusiasts alike, and is a good "gateway drug" for those having trouble getting into progressive metal.
Another one of my favorites off of this list. It's very much in the same vein as Number 7, with even some elements reused to great effect, but had a much more epic feel to it.
Delicious Spanish prog, which if you haven't heard before, you are in for a real treat. This band has yet to garner a lot of attention, and it's particularly difficult to get a hold of a physical copy of their first few albums, but III is probably their best so far and isn't as hard to find. Surprisingly, all their albums are on spotify.
Wobbler stems from the same vein as other nordic bands like Anglagard and Anekdoten, which is very clear from their first two albums. They took a different direction with this album, with a much heavier influence from 70s prog. Easily one of the best albums of the early 2010s so far.
Big Big Train - English Electric Part 1 & 2
This is technically two albums but they are meant to be played together as one, as themes carry over from one to the other in beautiful ways. This was Big Big Train's big big breakthrough. This album crowned them the kings of crossover prog. Their music is very approachable to both prog lovers and music lovers as a whole. This is one of those albums that I often show to someone in order to help them get into prog. There are lot of heavier prog elements combined with lighter and whimsical poppy elements, but never gets as sappy as most pop music does. Listening to this album is like taking a trip to the English countryside and paints a really vivid picture of it.
This band was pretty big in the 90s among prog circles, but they had fallen off the radar for a little while before this album was released. This album was a big comeback for them, and was similar to their earlier albums like Suffocating the Bloom and As the World. They have a lot of 70s prog and 80s neoprog influences, but have a very distinctive 90s alternative rock edge.
The Flower Kings - The Banks of Eden
This band has been around for a while, but this was their opus. This album has a lot of variety, ranging from the epic nature of symphonic prog to the weirdness of eclectic prog.
This album is fucking fantastic. It is the perfect blend of heavy prog combined together with jazz fusion elements all swished together in this huge double album chronicling the haunting adventures of a band of castaways who got lost of sea aboard their ship, the Unicorn, and slowly lost their minds and will likely blow yours as well by the end of it. No matter what types of prog you like or dislike, I'd recommend this album to anyone.
Steve Wilson - The Raven that Refused to Sing
I don't really have to say much about this one. Anyone that hasn't heard this and calls themselves a prog lover has been living under a rock. I may not be the biggest Steve Wilson fan here but this album sends chills down my spine. One of the most haunting and chilling prog rock albums that has ever been made.
An awesome follow up to their last album, Visions, heading in the same direction with even more classic prog influences, and to great effect. This album is super popular among prog circles and seems to pop up on this subreddit every few weeks, and with good reason.
This isn't their first album, nor their first well-known album. However, this marks the long awaited rebirth of Canterbury Scene. It is still has all the soft-then-loud jazzy elements and keyboard driven melodies that the old one did, but with a new contemporary edge.