Charismatic churches are characterized by their style of worship, with acts like speaking in tongues, holy healing services, being "slain by the Spirit," and so on. They are known to be very feel-good emotional and lively services.
The difference between Pentecostals and Charismatics is mostly over their denominational allegiance. Pentecostalism is its own denomination, and has its own doctrines and teachings beyond just charismaticism. Charismatics, on the other hand, can be found in just about every denomination (Baptist, Methodist, Catholic, Presbyterian, etc.) Another important difference is that Pentecostals see the manifestation of spiritual gifts, particularly speaking in tongues, to be evidence and a prerequisite that someone is saved. Charismatics are less likely to believe this, saying that it's evidence of being filled with the Holy Spirit, but regard it as a separate event from being born again.
Charismatic Christianity is one of the fasting growing traditions of Christianity in the world as well as one of the largest Protestant denominations.