The term Charismatic Pentecostal is derived from two biblical teachings. Let's go at it in the reverse order so you may fully understand.

Pentecostal is a name given to groups of believers who accept that there is a special event Jesus spoke of in the book of Acts. That is the Batism of the Holy Spirit. Acts 1:4-5 And being assembled together with them, He commanded them not to depart from Jerusalem, but to wait for the Promise of the Father, "which," He said, "you have heard from Me; for John truly baptized with water, but you shall be baptized with the Holy Spirit not many days from now."

We not only believe that Jesus commanded for His disciples to be baptized with the Holy Spirit then, but that He continues to do so today. The first disciples gathered together to wait for what Jesus had promised them and on the day of the Jewish celebration called 'Pentecost' the promise came true and all who were there praying for the promise received it. Today Jesus still freely gives to all who pray and ask for this same promise, the baptism of the Holy Spirit. Since the first baptism was fullfilled on the day of Pentecost those who receive the baptism of the Holy Spirit are given the name 'Pentecostal'.

Charismatic is the name given to those who believe that the Holy Spirit gives to Christians what is called 'Gifts of the Holy Spirit'. 1 Corinthians 12:4-11 There are diversities of gifts, but the same Spirit. There are differences of ministries, but the same Lord. And there are diversities of activities, but it is the same God who works all in all. But the manifestation of the Spirit is given to each one for the profit of all: for to one is given the word of wisdom through the Spirit, to another the word of knowledge through the same Spirit, to another faith by the same Spirit, to another gifts of healings by the same Spirit, to another the working of miracles, to another prophecy, to another discerning of spirits, to another different kinds of tongues, to another the interpretation of tongues. But one and the same Spirit works all these things, distributing to each one individually as He wills.

The word Charisma in the Greek is the word for gifts.

So in simple terms a Charismatic Pentecostal is someone who believes in the baptism of the Holy Spirit as a promised event from God and that the Holy Spirit gives supernatural gifts for the benefit of all of God's children.