The Catholic Charismatic Renewal has been around since at least the late 1960s. When I was growing up my mom was very involved in it, and I often attended prayer meetings with her. In the beginning it was sometimes dubbed the “Catholic Pentecostal” movement but later took the name “Charismatic” to avoid confusion with Protestant Pentecostals.
In a nutshell, Catholic charismatics believe that the gifts of the Spirit spoken of by St. Paul and mentioned in Acts, such as speaking in tongues, healing, prophecy, etc. still exist and can be manifested among Christian believers today. The charismatic style of worship tends toward exuberant praise similar to what you might see at a Protestant services. Sometimes (not always) this leads to liturgical abuse, but charismatic Masses can be done very reverently and are quite spiritually moving.
To some extent the charismatic movement in the U.S. peaked in the 70s and 80s, and many of the people I knew who used to be involved in it (including myself) now are more inclined toward traditional devotions such as Eucharistic adoration, the rosary, the Chaplet of Divine Mercy, etc. It does, however, remain very popular among Hispanic Catholics and in other parts of the world.
During the immediate aftermath of Vatican II, charismatic Catholics were often the most orthodox ones you would find. Unfortunately, some did leave the Church for Protestant churches they thought were more “Spirit filled.” Others became disillusioned with the overemphasis on spectacular charismatic “gifts” and a tendency toward Protestant concepts of what it meant to be a “born again Christian.”
Pope John Paul II endorsed the charismatic movement (though he did have some concerns about it) and Mother Angelica began her print and TV ministry in the 70s and early 80s as a charismatic – she even appeared on shows like the 700 Club and PTL when she first started EWTN.
If you tend to be a quiet, introverted type who prefers the Tridentine Mass, as a previous poster said, you probably won’t care for charismatic worship. If you are more extroverted or come from a Protestant or Pentecostal background, however, you might like it.