Small harquebuses and other handguns were carried by small numbers of light cavalrymen since the early Italian Wars. Jacopo di Porcia's The Preceptes of Warre written sometime in the early 1500s recommended a general employ some number of gunners on horseback, "For no sorte of souldyers, is more profytable thā they nor yet doth more myschife and hurte." Machiavelli in his Art of War also wanted a small number of his light cavalrymen to be armed with harquebuses, while the rest were armed with traditional weapons. The wheellock mechanism, which allowed a gun to be carried and used from horseback without constantly tending to a lit match, dates back to at least 1500 or so and some illustrations made by Leonardo da Vinci. The lock was a very delicate and difficult to manufacture device, but over the decades accounts of wheellocks and other self-igniting firearms in use gradually increase and the cost of these weapons was driven down.

The German Reiters themselves first appeared in companies sometime in the 1540s. They originally tended to consist of members of the lesser nobility who couldn't afford the full complement of a full man-at-arms and instead armed themselves with swords, various levels of armor ranging from mail or brigandine to 3/4ths plate and small harquebuses, petronels, or pistols, sometimes carrying up to 5 or 6 guns at a time. The exact role of the early reiters isn't entirely clear and may have varied. Some authors considered them another form light horsemen or mounted skirmishers while others gave them a more distinct, medium-cavalry role*. In either case the reiters proved very effective and soon made a name for themselves. At the battle of Saint-Vincent in 1552 massed reiters were eventually able to drive off the french cavalry, both light and heavy with their gunfire. At Saint-Quentin in 1557, deep squadrons of reiters were able to charge head first into the heavily armored French gendarmes and throw them into a panicked rout soon after the battle began.

*By the 17th century this would become more clear, with the majority of European cavalry fitting into one of two categories: the pistol-armed cuirassier, who took over the role of heavy cavalry as the lance fell out of use, and the harquebusier, who either skirmished from horseback or served in scouting, raiding, foraging, and various other light cavalry capacities. For some examples see No. 2 and No. 3 of this illustration

Despite these successes and a number of others, reiters and similar mounted pistoliers continued to have a somewhat mixed reputation for most of the 16th century. The French Huguenot captain Francois de la Noue wrote at length on the subject, blaming criticism of pistol-armed cavalry on a poor experience and the flaws of the increasingly popular "caracole" maneuver.

The caracole/fire-by-rank developed by reiters involved the first rank of a deep squadron of horsemen firing all their weapons once before withdrawing to the rear to reload while the second rank of retires stepped forward to repeat the process and maintain a constant hail of fire. This drill sometimes involved each rank making a brief charge as well in order to come within a short range before discharging their firearms while the main body of horsemen remained stationary at a distance. According to Thomas Digges, a very similar technique was associated with German men-at-arms and lancers as well, so it may even have predated the use of firearms somewhat.

The caracole may have been useful when skirmishing against another caracole or when attacking a stationary block of pikemen, it proved very vulnerable to a determined cavalry charge. A single reiter might be able to stay out of reach of a single lancer and continue to fire his pistols, but a dense squadron could not turn on a dime, and even if attempted it was too easy for a feigned retreat with many horsemen to turn into a real one.

. . . [when] they fetch about a great circuite either to charge a newe, or to take newe Pistols. Whereof it hath often come to passe, that they haue not had leisure so much as to turne their heads: for their turnes and returnes haue bene taken for a flight: wherevpon they haue bene so hotly pursued that they haue taken their carrier out right.

Similarly, shooting a single shot weapon from a galloping horse in combat was extremely difficult to do with any sort of accuracy. La Noue felt that a pistol under these conditions was unlikely to hit anything beyond 3 yards, even less than the reach of a lance! The solution, he explained, was that perfectly trained reiters would preserve their fire against heavy lancers until they were literally mixed with the enemy where pistols were far more terrifying and could pierce armor better than any lance, sword or mace. What's more, once the momentum of the charge is lost a lance loses all its power and becomes useless, while a pistol in melee can still fire in any direction with maximum force. From La Noue again:

although the squadron of Speares doe giue a va∣liant charge: yet can it worke no great effect: for at the onset it kil∣leth none, yea it is a miracle if any be slayne with the speare: onely it may wound some horse, and as for the shocke it is many times of small force, where the perfect Reistres doe neuer discharge their Pistols but in ioyning, and striking at hand they wound, ayming alwaies either at the face or thigh. . . Herein wee are to consider two things which experience hath confirmed: The one, that the Rei∣stres are neuer so daungerous as when they bee mingled with the enemie, for then be they all fire. The other, the two squadrons mee∣ting, they haue scarce discharged the second pistoll but either the one or the other turneth away.

Sir Roger Williams, writing in 1590 agreed in theory but stressed the high quality of pistolier was needed to be able to charge into a wall of lances. In his experiences with pistoliers

most do neither charge pistoll, nor enter a squadron as they shoulde, but commonly and lightly always they discharge their Pistols eight and fiue score off, and so wheele about; at which turnes the Launtiers charge them in the sides

So very disciplined reiters could sometimes beat lancers while lancers could usually beat reiters. Despite this, the use of lances continued to decline until they were almost completely gone from western europe at the end of the 16th century. Well-trained cuirassiers could now fulfill the same roles that the old heavy cavalry did while being more flexible and much cheaper.

I realize this may have gone a bit beyond your question, but hopefully i covered reiter-type cavalry, their roles and development, and the attitudes towards them in the 16th century.