I've been playing Medieval II: Total War a lot recently because it still holds up as a fun game after all this time. I was doing a throwaway game with my best bois Russia (incoming rant about how Russia are the best faction in spoilers)...
Okay, so it's not like they're perfect, but as the game goes they're pretty damn effective.
Pros:
Missile cavalry - Oh yes Russia sports cheap and plentiful numbers of the most overpowered troop type in the game. Kazaks can be made from the beginning of the game in large numbers and an army of them can spoil even late game stacks' day. Boyar Sons are a nice intermediate between combat cavalry and spear throwing cavalry (remember of course that spear throwing is still needlessly overpowered in M2:TW), Cossack Cavalry is the same, but with better combat potential, a real all purpose light cav. But even better than that....
Armoured Missile Cavalry - As anyone who hacked Rome Total War to play as Armenia knows, armoured cavalry archers are boss. They commonly have the defense of heavy cav, with the ranged capability of elite light cav. What's not to like? Dvor are (I think) the only such unit in the game, and as such bring a unique element that makes Russia stand out as a good faction.
Varied militia - Some factions (lookin' at you Scotland) have crappy militias. Worse, some have crappy militia varieties, making it hard for a city to self sufficiently defend itself. Russia sports a militia tree that goes; archer, spear, crossbow, cavalry respectively, meaning large cities can maintain powerful missile units, solid spear militia and mobile cavalry without intervention from a castle, making it easier to make money with Russian cities than other factions.
Dismounted Dvor - Wait, didn't you mention these before? Yes, but dismounted Dvor are worth the additional mention as all round excellent garrison material for large castles. They have the ranged punch of an elite archer unit (I'd place them about as rangey and deadly as longbowmen) while having the profile to take on combat duties as well. They won't outfight cavalry or dedicated heavy infantry, but as general purpose infantry these are an expensive but rock solid option.
Cossack Musketeers - I checked around before I established this, but this unit is the single highest damage gunpowder infantry unit in the game (18, the only one that comes close is Portuguese Arquebusiers at 17, but the Arquebus has less range. Normal muskets default to 16). They do the most damage at the longest range. The best part is that they become automatically available in huge cities from the walls alone, so no need to build the top tier barracks to get them! I've run off armies of feudal knights with these guys without them ever even reaching my gun line.
Orthodoxy - Russia is one of only two Orthodox factions in the game. This has the practical benefit that you don't get the Pope demanding you make peace with your enemies every five seconds, and you get the Ikoners Studio that gives you a public health bonus.
Cons:
Location - The geographical location of Novgorod means that it's hard to expand in the early game, and nigh guarantees you'll have an altercation with either the Mongols or Timurids. I lessen this by making my play for Scandinavia and Britain, making the impact of losing places like Smolensk and Riga less important for my overall survival.
Stack rating and micro - Remember all those missile cavalry armies I mentioned? The game thinks those suck. A full army of kazaks will always lose an autoresolve to even 3 or 4 units of dismounted feudal knights, so you have to micromanage every battle they fight. Now in the early game that means you're going to be playing out a lot of skirmishes, but it's kind of fun if you don't get bogged down into a war of attrition, which you shouldn't.
Orthodoxy - while it has its benefits, it also has its drawbacks. Being Orthodox means that you only have one faction that will look kindly upon you for your faith. Everywhere you go you need to convert people to your religion, and the catholic kings will look down on you if you go to war with their bretheren. Often though you can make common cause with distant Muslim Sultans in order to keep your alliance system in some semblance of order (your strength is determined as a combination of you and your allies by the AI).
Overall, I think they're one of the best factions in the game and enjoy playing as them very much. One added random factor is that when you can't see the majority of Southern Europe, the game tends to go a little weird. In my current game the Papal States have conquered all of Italy proper and Milan have taken all of what should be France. Hungary has taken the steppe villages I didn't care to take, and I took over all three Scandinavian countries and Britain. It's a weird old game...
...and my lands were invaded by the Timurids. Now the hordes in M2:TW suck, and I don't like them. They gave them several full stack armies and leaders who all uniformly have maximum dread. I know the hordes were a big deal but it tends to not work so well with the kinda wonky AI the game has. Either way, I'm prepared (more or less) and I have a strong army of dismounted boyar sons, spearmen and archers in a citadel armed with cannons to oppose them.
The siege begins well for me. The cannons kill their siege tower and scare some of their elephants into rampaging. I decide to pull my spearmen back behind my second line because their cavalry tend to fare poorly in urban fighting. But the very second my last spearman gets within the second line, a Timurid horseman follows him, captures the building and my second line is breached. To hold them I throw all four spearmen units into the gateway. The elephants they sent hilariously rampage their way through their own men and murder a lot of expensive heavy lancers, but my spearmen randomly keep breaking and running even as they're taking a heavy toll on the Timurid forces. At this point I've lost about 20% of my force and they've lost about 55% of theirs.
My boyar sons have been cooling their heels in the third line redoubt. I decide that it's time to relieve them and send out all but one of my boyar sons units, keeping the one with my captain within the final line with my last two units of archers (some of the others are still on the walls manning the guns, but aren't really doing much damage themselves). As they reach the spearmen on the front line, the spearmen break...as do all the boyar sons, to be ridden down without giving battle to the enemy at all. Great. I throw out the last unit in the hope it'll hold them back, only to see it flee as well and have nearly every unit I have cut down out of hand. The final stragglers are murdered by a tiny unit of about 8 mongol heavy lancers who managed to get through my last gate with the routing boyar sons and briefly took over the castle before my archers (remember them?) shot them down.
Now we have an impasse. I have two units of peasant archers. They have assorted remnants of halberd militia, and various horsemen units. Not a lot of them, but enough that they could kill these bow armed peasants in open combat. My third line is back intact, but they're bringing up a ram. I get lucky and a cannon shot annihilates the ram, and I sit while the turrets take care of the pushing militia. Now they only have one unbroken unit of halberd militia and no way to breach my walls. Usually that ends the game, except they still have rampaging elephant artillery outside the walls, which technically counts. Even though it doesn't.
So I send out my last unit with any ammo to shoot fire arrows at the last enemy to make them run away. I win!
Then several thousand other Timurids besiege my castle...