In the classical typology, you could think of dragons as Heavy Cavalry when operating on their own. This doesn't necessarily have much to do with horses, but rather it means that they are most useful as shock troops, using momentum to disperse enemy formations.
They also have very high mobility, and so can be used as Light Cavalry, pursuing the routed enemy troops. However, they share a couple of disadvantages with cavalry, namely:
- You can't very well use a dragon as a garrison.
- They are not very good at defence; if you let them get surrounded by a mob of enemy archers, they will either have to fly away or die a death of a thousand cuts.
- They can't be used for urban warfare (unless you want to just set the city on fire and level it, but that could destroy things you might have wanted to keep) or in difficult terrain, although the definition of difficult terrain is different for dragons and for cavalry. They probably won't be able to see what's going on under the forest canopy, in narrow gorges or anywhere there's overhead cover.
Additionally, compared to a squadron of Horse, a dragon can only be in one place at a time, so it might not be able to pursue an enemy that simply decides to flee in all directions.
The classical definition of dragoons is horse-mounted heavy infantry. They would ride somewhere on their horses, then dismount and fight as heavy infantry normally would. If you did just that, and took, say, spearmen for a mediaeval equivalent, you could have dragon-mounted heavy infantry.
As mentioned in Dan's answer, you could use them in some combined arms tactic; e.g. have the dragon drop your infantrymen, have them form a line and advance against a similar formation of the enemy, only having the dragon use its fire-breath on either the enemy line of battle or their archers before the lines meet. This would amplify the effectiveness of both kinds of troops, and allow them to succesfully engage the enemy in greater numbers than they could separately.
You could also have the infantrymen garrison a (possibly makeshift) fortification and use the advantage of air-support (and effective counter-artillery) to defend a strategic spot, while the dragon could retire to a covered courtyard to rest.
You could also try assaulting a city by having the dragon create a wall breach (either by fire-breath or just by dropping rocks) and then storming the breach, and continuing the fighting inside. This city assault would be much more effective than the traditional way, because you can just fly in and attack immediately, without having to wait for siege engines to be built or hauled in, and giving the enemy very little notice.
If we abstract the definition of dragoon a bit, we could also try mounting other kinds of troops on dragonback.
Archers might come to mind, but I'd be disinclined to try that; they won't do much more damage than the dragon itself on foot, and it would be extremely difficult to fire accurately from dragonback.
So, you could use light infantry. They could penetrate forests and other terrains with overhead cover, and try to drive the enemy out from under it, where the dragon could incinerate them. Having a dragon around would also probably discourage enemy heavy infantry from forming up, so you light infantry could move and advance with impunity in open terrain (if no cavalry is present or if dragons spook horses).
If you're really into high mobility and can manage the technical difficulties, you could try strapping horses onto a dragon and have him transport a unit of light cavalry. They could seek isolated battalion-size units of heavy infantry, have the dragon attack them to break their formation and then use both the dragon and the cavalry in pursuit, for maximum casualties.
Depending on your setting and what kind of troops (and training for them) you have available, using a dragon as a transport will significantly broaden your tactical options. You could have a unit of either heavy foot, light foot, dual-trained troops (though these would be considered "elite" and therefore rare and expensive) or even cavalry transported on dragonback, and deployed tactically for best effect. If you can manage it logistically, dragon-mounted dragoons are definitely a viable option.
By the way, this has sort-of been explored in literature; Naomi Novik's Temeraire series describes dragon combat in the Napoleonic era. Some dragons do have breath capabilities, and these are used primarily as shock troops (or against ships, which is strategically crucial in the setting), the others mostly use direct physical attacks against ground troops or other dragons, but also have a platoon of riflemen on their backs. Because dragons are bonded with their captains, mid-air boarding is also a common tactic. Applications of dragons as strategic transport are also explored in the series.