As a Czech native, we prefer to call it the Hussite Revolution, where the war "Hussite Wars" were a by-product accompanying the revolution. It was a true revolution as would be described in the book, The anatomy of revolution by C. Brinton. The revolution wanted to change the medieval order and implement a better system in the era, when the late medieval Bohemian/Czech society experienced a significant social, cultural, ethnic, demographic, economic, and political crisis.

The root of the revolution was seeded in the late 14th century when extremely wealthy, prosperous, and peaceful (Bohemia has not seen a war between 1310 and 1396) experienced a significant shakedown of its medieval institutions accompanied by a dynastic dispute, indecisive rule of the king Václav IV, schism, and economic crisis following the outbreak of the Black Death around 1380 and returning again 24 years later (Bohemia was spared of the Black Death in 1340s). While Václav IV is negatively looked in the German world, he remain(ed) relatively popular within Bohemia across the social classes, while is more ambitious step-brother and Hungarian king Zikmund was not, due the fact he looted the Bohemian treasury in Kutna Hora and caused a devaluation of the stable coinage, impoverishing many.

The Hussite Revolution was primary religious upheaval caused by the critics of the Catholic church coming from the church itself.John Huss was the most outspoken critic of the church and people identified the existing church hierarchy as the primary cause of the ongoing social, political, and economic crisis that fully exploded in the 1410-1418. The church was widely attacked by various social classes, magnates, city burghers, and the university for failing its mission. The church in Bohemia was also represented by archbishops like Jan z Jenstejna who instead working with the executive power went directly with challenging the royal authority. Václav IV, while rather a weak king, remained in power for 40 years, prevented full blown civil war between the Papal forces, conservatives, and Zikmund on one side, and the reform minded urban population led by Prague, the knights, and the university. The death of the king unleashed the Hussite Revolution.

Bohemian estate refused to acknowledge Zikmund as the king as he was discredited from humiliating the king, he did nothing to prevent execution of J. Huss, and he was unwilling to deal with ambitious warrior knights as an equal opponent. In order to secure the crown and enforce his rule over Bohemia, Zikmund led the I. crusading war against Prague where he was defeated in 1420. This led to widespread war lasting for another 14 years. The Hussite forces were represented by several military units (we called Prague military pact, Orebites in East Bohemia, and Tabor in southern Bohemia) that were unified when it comes to the external threat, but fought for supremacy within Bohemia. The catholic forces within Bohemia disintegrated by 1424. The catholic administration caesed to exist as the abbeys, parish churches, and collegial institutions were either physically demolished, or were taken by the Hussite priesthood.

The Catholic forces launched five crusades against Bohemia ( 1420, 1421, 1422, 1427, 1431), but without success. The war also spread out to neighboring countries, where the Hussite army punished the lands that supplied the forces for the crusade. Between the Hussites also came a deep ideological split caused by a different social position, where moderate wanted to end the war, while the radicals wanted to fight. The primary source of the dissatisfaction was Pilsen. Pilsen was originally a Hussite town, but sided against them. It was the only main Czech speaking city that remained forever loyal to the Catholic cause. Pilsen also became the center of the Czech Catholicism in years, when 85% of the population were the follower of the Hussite faiths and its splinter groups (Bohemian and Moravian Brethren and so on). The failure of the radicals to subdue Pilsen caused them to lose the war in 1434. However, the power of the Catholic church was broken.

The revolution ended the role of the Catholic church within Bohemia. The Catholic administration lost its archbishopric which was vacant till 1541. The bishopric of Litomysl was permanently closed. The majority of the abbeys was destroyed and ceased to exist. Prague and primary cities in Bohemia came as a the winner of/from the Revolution. The wealthy magnates and nobility also gained from the revolution. The social inequality of the 14th century was bit erased and the 15th century was looked as era of price stability. The big loser of the revolution and the wars was Bohemian culture, peasantry, and the knights. The Hussite era in Bohemia was closed with the death of king George of Podebrady (+1471).

The Czechs are not unified on the analysis of the Hussite Era, and you can read various materials that prize the military achievement and the warrior spirit of that time, while many would condemn senseless killing, mass slaughter of real and perceived enemies of the revolution, and the destruction of the Bohemian wealth and culture. I personally do not have a favorite author lets be Cornej and his books about Hussites or Hussite Revolution by Bartos. I personally like the civilized discussion on there: Husitstvi where individuals put a lot of information about specific battles, warriors, and the time.

u/kaik1914 addressed the important political and social issues involved in the Hussite Wars, so I’ll try to get at some of the religious issues and a bit about Jan Hus himself.

For context, we need to go back more than a century. It's all related, I promise. In Hus’ time, the Catholic Church was in the midst of the Western Schism. Back around the turn of the 14th century, Pope Boniface VIII had been involved in a years-long disagreement with Philip IV of France over whether the monarch had the right to tax the church and appoint church officials in France. The argument went back and forth for quite some time, with Philip seizing Church funds, Boniface excommunicating Philip, Philip spreading rumors that the Pope kept mistresses, and so on. You know, good fun. Boniface finally pushed Philip too far in 1303 when he declared that only he (the Pope) had the right to fill any vacancy in French churches, demanded that Philip come to Rome to answer for himself, and excommunicated him a second time. Philip responded by sending soldiers to Rome, where they broke into Boniface’s palace, arrested him, and threw the 73 year old Pope into a cell where he was held and beaten for three days. He was released, but died a month later. In the aftermath of these events, Pope Clement V moved the Papacy to the city of Avignon in southern France, where it remained, under French influence, for the next several decades.

Fast forward to 1378. Pope Gregory XI died. A Frenchman, he had been visiting Rome for the past year, but had announced his intention to return to Avignon. The College of (mostly French) Cardinals convened in Rome to elect a new Pope. While they were meeting, an angry mob of Romans demanded that the cardinals choose an Italian who would move the Papacy back to Rome permanently. Fearful of the mob, the cardinals quickly elected Urban VI, an Italian. Urban settled in Rome, while the cardinals quickly left the city. Urban soon disappointed the cardinals, and a majority of them met again a few months later. They declared that Urban VI was not legitimately elected, since he was only chosen out of fear. The cardinals then elected another Frenchman, who took the name Clement VII, declaring he was the true Pope. Clement took up residence in Avignon.

Now, there had been antipopes before. There had always been people who claimed the papacy without any real legitimacy. A man is only legitimately the Pope if he is chosen by the College of Cardinals. But now both men were chosen by the same body of legitimate electors. Who was the true Pope? Some cardinals remained loyal to Urban. Both supposed popes appointed new cardinals to their own colleges. When Urban died, the Roman cardinals convened and elected Boniface IX. When Clement died, the Avignon cardinals met and elected Benedict XIII. Both lines continued to appoint new cardinals and elect new popes. Meanwhile the various countries of Europe had to decide which claimant to support, which fit into ongoing rivalries and wars. France supported Avignon, of course, so naturally England backed Rome. Scotland then, unsurprisingly, supported Avignon, and so on.

It was in this context of clergymen in fancy robes and luxurious palaces, squabbling over who was the real pope like children on a playground, that Jan Hus appeared on the scene. Like a great many people, Hus was disgusted by the schism. Following the earlier English theologian John Wycliffe, Hus saw no biblical basis for the institution of the papacy. He taught that a person could be justified in the eyes of God and go to heaven even without being subject to the Pope. This was directly contrary to the Papal bull Unam Sanctam, issued by Boniface VIII in 1302 at the height of his conflict with Philip IV. Boniface had declared that “Outside of the Church there is no salvation,” and that “it is entirely necessary for salvation that every human creature be subject to the Roman Pontiff.” Hus also wrote that nobody, not even the Pope, had any right to spread Christianity by the sword. Crusades, therefore, were unchristian and illegitimate.

Hus further taught that indulgences had no power and should not be sold. This position was popular among the common people of Prague, because the Church had raised vast sums of money by selling indulgences to them. Hus even gained the (tentative) support of the King of Bohemia. But by denying Papal legitimacy and arguing against indulgences, Hus threatened both the Pope’s power and his revenue stream. In response, the Pope (the one in Rome) pronounced an interdict over Prague, banning all church services and sacraments. That meant no marriages could legally take place, nor could priests perform baptisms, hear confessions, or distribute communion. Hus voluntarily left the city, but the conflict continued to grow.

Meanwhile, a Church council convened in the city of Constance in southern Germany beginning in 1414. Called by the emperor-elect of the Holy Roman Empire (incidentally, the brother of the King of Bohemia), its purpose was to find a solution and put an end to the schism. An earlier council had met in Pisa in 1409, but while it had succeeded in electing a new successor to the papacy, neither of the two existing claimants accepted it, so now there were three popes. Hus was called to the Council of Constance. He was promised safe passage, but was arrested as soon as he arrived in 1415. He was offered a chance to recant all of his teachings, but refused. He was condemned as a heretic and burned at the stake.

Hus’ followers in Bohemia were furious. The King ceased even pretending to comply with demands from Rome. The Council of Constance meanwhile succeeded in deposing the rival popes or convincing them to resign, and elected Martin V in 1417 as a compromise candidate, thereby ending the schism. Violence broke out between Catholics and Hussites in Prague in 1419, and in 1420 Martin declared a crusade against the Hussites. He issued the bull Omnium Plasmatoris Domini, which declared that all followers of Hus were heretics and authorized killing them. A series of invasions and wars followed over the next 15 years or so, none of which was truly successful. These are what are referred to as the Hussite Wars. For more detail on the course of them, see the answer by u/kaik1914.

As for whether they were successful or not, from the Catholic perspective in Rome they were a failure. The Hussites were not defeated, exterminated, or converted back. Hussites remained a majority in Bohemia for centuries. It was only in the 1620s, in the opening stages of the 30 Years’ War, that the Holy Roman Emperor regained control of Bohemia. The Habsburg monarchy (and later Austrian Empire, later still Austria-Hungary) persecuted the Hussites and tried to force them back to Catholicism. This campaign largely ended in the 20th century with the breakup of Austria-Hungary, but by then membership in traditionally Hussite churches constituted a small minority in the region.