Phew big question, and a good on. I shall try and keep my answer to a reasonable length so please forgive some generalisations but I will include some recommended reading at the end, and of course if you want to pick up on anything here afterward then just ask away.
A common habit when we look at types of military units throughout history is to consider only their battlefield use - this is understandable and, if your historical interests focus on battles, natural. However, cavalry - particularly during the early modern period actually did most of its work outside of battles. In fact, it was this period that saw the shift from cavalry as the primary power on the battlefield, to that being infantry (although this had been in flux throughout history and differed across periods and armies). However, the early modern period in Europe saw a massive explosion in army size as state capacity, technology, tactical battlefield developments, and strategic changes, as well as broader organisation all came together to change the dynamic of warfare to an almost unrecognisable scope. In 1550 France could put an army of around 20,000 soldiers in the field, by 1700 France's (multiple) armies fielded perhaps 500,000 soldiers.
For the cavalryman of the early modern period, this shift in army size dramatically changed their use. No longer was cavalry needed to charge and break enemy infantry formation therefore heavily armoured prestige (heavy cavalry) or cheaper units focused to pursue broken unit or chase away poorly equipped infantry such as archers (light cavalry). Well-drilled infantry, now wielded in large tactical blocks, guarded by pike and backed up with muskets was less vulnerable to cavalry and indeed could be quite deadly to cavalry themselves if they got close. Instead, cavalry began to take on ever more increasingly essential non-battlefield roles. These types of roles had historically been performed by light cavalry, with heavy cavalry saved for the hammer of the battle, but the costs of fielded large forces, and subsequent demands on cavalry work, meant that there was no longer "battle only" cavalry, instead it had to be willing and able to perform more roles - a jack-of-all-trades. Scouting in force, with enough strength to drive off enemy scouts as well as protecting the increasingly long and extensive lines of march of the huge numbers of infantry and accompanying artillery and baggage trains took up a lot of cavalry even if the infantry and artillery were stationary. However, breaking into smaller units and ranging days ride away from the main army to source, gather, and escort vast amount of food and drink from civilians to support the army was another. Just to consider that last one, some of these cavalry units had to be several days away from the army, operating in hostile territory, with enough force and numbers to be able to defend themselves from enemy scouts/foragers, as well as to ensure the civilians supplied them without resistance, and be aware enough of the armies movements to be able to send that day's forage to where the army WOULD be by the time the supplies reached the army. Meanwhile their role on the battlefield continued - but now it was to meet and combat the enemy cavalry, hopefully neutralise it, so then the victorious cavalry could support their own attack on the enemy. Cavalry became less able to be the battle-winning charge in infantry-heavy battles simply because the infantry had learned how to counter him. Just as the infantry needed the cavalry to keep away enemy cavalry, and to help guard and supply his line of march, so too now did the cavalry need the infantry to win a battle.
All this meant that the early modern cavalryman needed to be able to fight (and win) in a cavalry on or off the battlefield, but also range far ahead, quickly, of the army, and to reliably make use of his speed and flexibility for all his attendant roles. His armour and equipment, even his horse, all changed to meet these needs and the end result - often called the harquebusier - was a moderately armoured, usually in leather rather than plate, heavily equipped with at least two pistols, a carbine, a strong sword, and perhaps a dagger. They were also a well-mounted warrior, capable of holding his own on the battlefield while also performing all the other onerous, dangerous, less prestigious tasks that his role now demanded of him. His horse was in many ways bred to the purpose, a knights charger was too heavy, too slow, whereas a light cavalryman's mount was too light for the melee and to carry the equipped trooper. Cavalry mounts of the early modern period had to be reasonably tall, strong, but fast as well. But the harquebusier's name gives a hint to the newer development that would begin to change cavalry forever - the harquebus, or firearm. Most cavalry from this point on began to be equipped with some form of carbine, and as technology evolved the cavalry drew priority for the best firing mechanisms - even in the 1640s, when an infantryman would wield his matchlock musket, the cavalryman would be equipped with a flintlock carbine. This was primarily used for dismounted work, turning the cavalry man of the early modern period into a sort of heavily armoured dragoon when need demanded. This need for mobile firepower would be supplemented by actual dragoon forces in the ensuing centuries, with more cavalry forces gradually being turned into dragoon regiments as the need for mobile and flexible firepower, alongside spending demands, triumphed over that of melee combat.

TLDR: They substantially changed in form - armour, equipment, and mount - all to meet the requirements of their changing and expanded roles off and on the battlefield.

Further reading
Primary sources on cavalry and their use:

Markham, G., The Second Part of the Soldiers Grammar: or a Schoole for Young Soldiers Especially for All Such as Are Called to Any Place, or Office (How High or Low Soeuer) Either in the Citie, or Countrey, for the Training, and Exercising of the Trayned Band, Whether They Be Foote or Horse. Together vvith Perfect Figures and Demonstrations for Attaining the Knowledge of all Manner of Imbattailings, and Other Exercises (London: 1627)

Turner, J., Pallas Armata: Military Essayes of the Ancient Grecian, Roman, and Modern Art of War. Written in the Years 1670 and 1671 (London: 1683)

Secondary
Something secondary covering the changes of warfare:

Fissel, M.C., English Warfare 1511–1642 (London: Routledge, 2001)

For an overview on the academic story of the development of warfare in the period see:

Rogers, C.J. (ed.), The Military Revolution Debate, Readings on the Military Transformation of Early Modern Europe (Boulder, Colorado: Westview, 1995)