What effect did the Reformation have inside the Catholic Church?
The Protestant Reformation got on it way when Martin Luther posted his Ninety-five Theses (Disputation on the Power and Efficacy of Indulgences) on the door of All Saints' Church and other churches in Wittenberg, in accordance with University custom, on 31 October 31, 1517.
31 October 1517, the day Luther sent the Theses to Albert, was commemorated as the beginning of the Reformation as early as 1527, when Luther and his friends raised a glass of beer to commemorate the "trampling out of indulgences". The posting of the Theses was established in the historiography of the Reformation as the beginning of the movement by Philip Melanchthon in his 1548 Historia de vita et actis Lutheri. - Ninety-five Theses
In large, this lead to the Catholic Reformation from 1545-1648. Thus this is the time frame, that interests this question matter the most, but not exclusively.
The Counter-Reformation, also called the Catholic Reformation or the Catholic Revival, was the period of Catholic resurgence that was initiated in response to the Protestant Reformation. It began with the Council of Trent (1545–1563) and largely ended with the conclusion of the European wars of religion in 1648. Initiated to address the effects of the Protestant Reformation, the Counter-Reformation was a comprehensive effort composed of apologetic and polemical documents and ecclesiastical configuration as decreed by the Council of Trent. - Counter-Reformation
So what effect did the Reformation actually have inside the Catholic Church?
The Council of Trent will take on the largest portion of how the Reformation affected the Church with her understandings of her doctrine, liturgy and evangelization in the missions and in Europe, as we can see below:
The council, in the Canon of Trent, officially accepted the Vulgate listing of the Old Testament Bible, which included the deuterocanonical works (called apocrypha by Protestants) on a par with the 39 books found in the Masoretic Text. This reaffirmed the previous Council of Rome and Synods of Carthage (both held in the 4th century AD), which had affirmed the Deuterocanon as scripture. The council also commissioned the Roman Catechism, which served as authoritative Church teaching until the Catechism of the Catholic Church (1992). - [Counter-Reformation
Priestly formation in Catholic seminaries was greatly improved through the Council of Trent, thus trying to see to it that the clergy, both priests and bishops were properly trained in regards to doctrine and the administration of the sacraments. Many members of the clergy were teaching seriously erroneous view about doctrine prior to this, especially in the domain of the sacraments, indulgences, relics and purgatory to name a few.
Such reforms included the foundation of seminaries for the proper training of priests in the spiritual life and the theological traditions of the Church, the reform of religious life by returning orders to their spiritual foundations, and new spiritual movements focusing on the devotional life and a personal relationship with Christ, including the Spanish mystics and the French school of spirituality.
Parish priests were to be better educated in matters of theology and apologetics, while Papal authorities sought to educate the faithful about the meaning, nature and value of art and liturgy, particularly in monastic churches (Reformed Protestants had criticised them as "distracting"). Notebooks and handbooks became more common, describing how to be good priests and confessors. - Council of Trent
Both clergy and laity were slowly better equipped to learn their holy truths with the publication of the Roman Catechism. This catechism is still in use in more traditional Catholic circles.
The Roman Catechism (or Catechism of the Council of Trent, published 1566) was commissioned during the Catholic Counter-Reformation by the Council of Trent, to expound doctrine and to improve the theological understanding of the clergy. It differs from other summaries of Christian doctrine for the instruction of the people in two points: it is primarily intended for priests having care of souls (ad parochos), and it enjoyed an authority within the Catholic Church equalled by no other catechism until the Catechism of the Catholic Church (1992). The need of a popular authoritative manual arose from a lack of systematic knowledge among pre-Reformation clergy and the concomitant neglect of religious instruction among the faithful. - Roman Catechism
In 1570, Pope St. Pius V publish his now famous bull Quo primum, made the Roman Missal, as revised by him, obligatory throughout the Latin Rite, except for those places and congregations whose distinct rites could demonstrate an antiquity of two hundred years or more. Thus Pope St. Pius V united the whole Western Catholic Church under a single Roman Liturgy of the mass.
Let us also recall that the printing press aid both sides of the Reformation and the Counter Reformation. Bible reading became slowly more accessible to many as illiteracy decreased.
The printing press
After the invention of the printing press, prior to Luther's Bible being published in German, there had been over 20 versions of the whole Bible translated into the various German dialects (High and Low) by Catholics. Similarly, there were several vernacular versions of the Bible published in other languages both before and after the Reformation. The Church did condemn certain vernacular translations because of what it felt were bad translations and anti-Catholic notes (vernacular means native to a region or country).
The Catholic Douay-Rheims version of the whole Bible in English was translated from the Latin Vulgate. It was completed in 1610, one year before the King James Version was published. The New Testament had been published in 1582 and was one of the sources used by the KJV translators.
The Latin Vulgate was always available to anyone who wanted to read it without restriction. Some Evangelicals have said that it would only have been usable by people who read Latin. But in the 16th Century there were no public schools and literacy was not that common, especially among the peasants.
The foreign missions during this time flourished and Religious Orders deepened their spirituality and commitment to the Church.
To give just one example, for the efforts done in the area of evangelization, I will let Dr. Warren H. Carroll Ph.D. In his book The Cleaving of Christendom  writes the following on page 626:
The conversion of Mexico was by far the greatest and most complete in all missionary history, and Mexico remains one of the most Catholic countries in the world today. And most of that conversion took place in the years from 1532 to 1536 - the very years when Henry VIII was taking England out of the Church. What the Church lost in the Old World, she regained in the New.
St. Theresa of Avila reformed the Carmelite Order in Spain, which spread to many other countries.
The Carmelite Reform
As time went on, problems caused laxity among many religious Orders and reforms were needed. Carmel was no exception and many tried to bring about reform among different houses. It belonged to the unique genius of St. Teresa of Jesus (of Avila), our 16th century Spanish Carmelite reformer, to recapture the spirit of the original Rule of Carmel. Inspired by a deep love for Jesus and a strong desire to help His Church which was undergoing turbulent changes in that era, she began to establish new communities within Carmel with a vision to live Carmelite life in the spirit that the first hermits of Mount Carmel lived it.
In 1562, she established the monastery of St. Joseph, the first monastery of her reform in Avila, Spain. She continued to form small communities of nuns (originally, not more than 13 in each house) who were totally dedicated to prayer and sacrifice for the Church.
Her communities became known as Discalced Carmelites. Discalced means barefoot, a sign of reform at that time. With the collaboration of St. John of the Cross, her reform later extended to founding communities of men. Between 1567 and 1582, St. Teresa founded 17 monasteries for nuns and 15 monasteries for friars. She died in her convent in Alba de Tormes in 1582 and was canonized in 1622 by Gregory XV. On September 27, 1970, Pope Paul VI declared her a Doctor of the Church.