There isn't exactly a clear line of separation between "classic" and "modern" shooters, but if I had to choose one, it would be 2001's Halo: Combat Evolved. (Other people would probably choose Half Life). In any case, Halo clearly marks the beginning of the popularity of the so-callled "military shooter." Before this point, it was mostly about being a lone fighter, fending off countless enemies. But now suddenly, you're a part of a larger fighting force, a military. Again as I said, there is no clear line of separation, so this game definitely still holds onto some shooter staples of the past. However, the things it brought to the table have stayed with the genre (especially on consoles) into today. If you are into console FPS, this game should mean a lot to you. Basically, Halo picked up the torch where Perfect Dark, another influential console shooter, left off.
First of all, is the weapon system. FPS games going back to Wolfenstein 3D have had limited ammo, but Halo adds a new limited resource in the form of guns. See, you can only hold two weapons at a time. This probably doesn't sound unusual to you, because most modern FPS games have this setup, but before this you were a one-man walking armory. You could carry lots of weapons in hammerspace. Halo, with this limit, took a step towards a more realistic weapon system. However, ammo is still limited. This means you'll constantly be scavenging dead bodies weapons, especially when you run out of ammo.
Speaking of weapons, there's a large variety on display here. On one hand, you have standard FPS staples -- Pistols (one of the most powerful pistols in all of gaming), assault rifles, shotguns, turrets, sniper rifles, and so-on. More interesting are the alien weapons, which are by and large less desirable than the human ones, but each have an interesting gimmick. The plasma pistol can be charged, the plasma rifle overheats but fires quickly, the Needler fires homing shots that attach and then explode. It gives you pretty nice variety of ways to deal out carnage. Each weapon has advantages and disadvantages, especially against certain kinds of enemies. For example, you can take out an enemy's shield with a fully charged shot from the plasma pistol. You've also got grenades that you can throw at any time (uncommon in FPS games at the time), which come in two varieties - Regular frag grenade, and plasma grenades, which stick to opponents. Plasma grenades are the standard grenade of the enemies, and if one gets stuck to you, that's it, you're dead.
There is also vehicles, something that this game pioneered, but they've aged badly. Vehicles like the Warthog are great for cooperative play, but the AI soldiers are notoriously bad at shooting at what you'd like them to. Much better are vehicles where you have control of the weapons, like the alien's Ghost vehicle. However, the steering controls are pretty awful. In order to turn the vehicle, you turn the camera and the vehicle turns with it. However, there's a horrendous delay and it makes driving a nightmare. Every Halo game in the series has this same problem and I hate it. The vehicles, though, feel like they've actually got weight to them, so going off of jumps feels pretty satisfying. Though it does mean you can flip your vehicle, so there's a big reward for careful driving.
However, the ground controls set the standard for console first person shooters. It seems obvious now -- The left stick moves your character, the right stick turns you and looks up/down, the right trigger fires your gun as if pulling the trigger of a gun. Before this, console FPS controls were all over the place, especially before the second analog stick. There is some auto-aim to assist you, but a console FPS without that would be a terrible thing. Overall, the controls have aged perfectly.
The enemies offer an interesting variety as well. You've got Grunts - the basic enemies forces, Elites - leaders and tougher to take out, Hunters - walking tanks who are best attacked from behind, and so-on. Coupled with the weapons, this makes for very interesting encounters. As stated previously, certain weapons will work better on certain enemies, but the enemy AI itself is done pretty well. Enemies rarely just charge at you. A great example is the Grunts -- they like to have an Elite around. If you kill their Elite, they will panic and start to run. Elites themselves duck and dodge your attacks. The sword wielding ones are especially pretty sneaky.
Halo had more emphasis on cinematic story than a lot of FPS games of the time, featuring some of the best cutscenes of the era. These games sure had come a long way from the text-crawls from Doom. Interesting to me, is how well this game manages to shift tone. Starting off, it's a sorta sci-fi adventure military story. The setup is standard - The humans are in conflict with an alien entity known as The Covenant, named so because it features different species of beings. You are a super soldier, The Master Cheif, fighting for the UNSC - The United Nations Space Command. The Halo series actually has a lot lore behind it, but for most of it you have to learn about it through non-game materials (books, movies, etc). Later in the game, the tone shifts to horror, and even later to a desperate struggle to save the universe. Everything is fleshed out pretty well, it's not just "good vs bad." The game takes place on a giant ring-shaped world called Halo, which as it turns out, is a giant super weapon built to destroy all sentient life in the galaxy. The Covenant want to activate Halo because their religion dictates that they should, the UNSC want to stop The Covenant, and, without spoilers, a third party wants to fire off Halo because, well, that's just his job.
The level design of Halo is probably the thing that has aged the worst. Don't get me wrong, the levels all look fantastic for the time. There's thick foliage in places, and the indoor areas of Halo are big and open and feel totally alien. The levels' design set the mood of that point in the story perfect. Early levels feel like going on an adventure, the horror segments are darker and often claustrophobic. However, there is ton of re-use of set pieces and room design. Often, you will find yourself in a room that is totally similar to rooms you've been in countless times before. Not only that, but the whole game feels like a trek to point A to point B back to point A. I don't mind this too much, but it begins to really feel like they're stretching out the length of the game.
There was a lot of hype built up around Halo when it was first released, but I don't think it's the most amazing game ever. But, it was very influential to the genre, and still a damn good time. I highly recommend playing it. I can't say that as much with the sequels, but that's a story for another time.