As far as what warhorses were active during the middle ages it's not the big drafts like the one I have in the following pictures. They were smaller but stocky enough to carry all the armor the horse had to wear as well as the Knight's. They were known as Destriers.
The Destrier was a highly trained warhorse. They were as much a warrior as the Knight himself. The drafts of today were really the pulling horses even back in the Dark/Middle ages. I get a lot of crap for riding a draft horse in the medieval society because some people are what we call "period nazi's" and believe you must recreate to the exact specifications.
Palfries were the riding horses most of the nobles would have these horses. Coursers were the cavalry horses. Very fast, very agile. Rouncies were the all purpose stock horses. Pulled wagons, peasants rode them, they are what we call grade horses today.
And remember, breeding wasn't really known back then like it is now. They bred for the work the horse needed to do but for someone to try for TB's or QH's like today, they just didn't do that back then.
Along with the horses there were definitely 'warhounds' used. They could provide excellent killing blows from below and the enemy often didn't expect them. They were trained to go for throat, groin, legs - anything to bring the enemy down so they kill them.
I belong to an organization known as the SCA - Society for Creative Anachronism. It is a world-wide organization. Basically you pick a time prior to 1650 A.D. And you recreate it fully. From the clothing to the camping to the cooking to the writing to everything in between.
My chosen time period is 145 A.D., Greek/Roman. So my armor is in the Greek/Roman style. I do swordfighting. We use rattan swords because while we do hit each other you don't want to actually get killed so no sharp edges. And these guys hit hard!
There are tournmanets and wars. Tournaments are individual fighting. Each part of the world is divided up into Kingdoms and those Kingdoms are divided up into Principalities, Baronies and Shires. There is a King and Queen of each Kingdom and a Crown Tournament is held 2-3 times a year to fight for the Crown. The winner of the Crown Tournment becomes King/Queen and rules for a 3-6 month period.
The medieval horse part of the organization is the same way. You recreate your horse armor/barding in the fashion of the time period that you choose. Because I'm early period, Solon doesn't have anything in the way of armor or barding. Although I have toyed with the idea of making a four horned Roman saddle.
Our medieval games included the Rings (where you ride straight down a line and pick up the Rings with your lance that are hanging from "T" piec of wood, there is the Quintan which you take your lance and slam it into a a wood piece and however many times it spins around determines the winner (only you don't want to get hit in the back of the head with the sandbag as you go by!), there is the game of Heads where you do a pole bending pattern and knock the heads off the pole with your sword.
There are other games but these are the main ones.
Here are some pictures that kind of show what I'm talking about (captions above the pictures):
This is my full fighting kit. I do have a new helm which I will show below:
This is my new helm - it's super heavy but very protective. I had my armorer design two horses in the grill facing away from each other:
This is a nasty bruise I received after a sword practice and that went through my leather armor and padding underneath! You can see the outline of the sword blade from top to bottom.
Here is a picture of one of our King's. He's been King more than once. The white belt denotes that he is a Knight.
The medieval games. Solon was just a baby at this event (just turned 4 and I had only started riding him a couple of month before the event so we did the games only at the walk).
The Quintain - on the other side other wood shield is the sandbag and if you don't get past it fast enough while going at the trot/canter it can knock you in the back of the head.
The Rings - not so easy to do even at the walk!
The Heads - basically a pole bending pattern that you knock the heads off with the sword as you go by.
Sadly, Solon grew out of the horse trailer I had so he retired as medieval warhorse.
We don't do 'real' jousting due to insurance issues. They do what is called boffer jousting which in my opinion is a joke. It's a styrofoam lance.
If you are interested in learning more about the SCA and how to find a group next to you here is the link.
Society for Creative Anachronism, Inc.
This link will show you how to find your local group:
SCA, Inc. | SCA Geography