Recently, I acquired the domain CharismaticCalvinist.com and several friends asked me to explain this seemingly oxymoronic combination of words.
The term "Charismatic Calvinist" can probably be attributed to Pastor Mark Driscoll of Mars Hills Church who associated the term with himself at a pastors conference in 2005 according to this wiki article. Another term used to describe this position is Neo-Calvinism or Neo-Reformed. According to one Time's Article, one would get the impression that this is simply Calvinism paired with contemporary Christian music; it more complex than that and frankly, I personally can't stand most contemporary (pop) Christian music, but I digress.
Charismatic Calvinist combines two theological systems of thought. The first is Calvinism, a theology taught by John Calvin which emphasizes a high view of Scripture and the sovereignty of God from which it forms its theological views commonly described by the acronym T.U.L.I.P meaning:
Total Depravity (a.k.a. Total Inability and Original Sin)
Unconditional Election
Limited Atonement (a.k.a. Particular Atonement)
Irresistible Grace
Perseverance of the Saints (a.k.a. Once Saved Always Saved)
The details of Calvinism are beyond the scope of this post but if interested you can read about it here, and here.
Regarding the second component to Charismatic Calvinism, I am NOT speaking about that Pentecostal flavor of Charismatic thought that believes that the baptism of the Holy Spirit is a second work of grace that brings power in the life of the believer evidenced by the speaking of tongues. I am also NOT speaking about the charismatic 
movement having all those bizarre elements like raising of the dead, spiritual barking and laughter, crawling and shaking on the floor, future-telling and other counterfeit nonsense. Side note: a good book to have in hand is Counterfeit Revival by Hank Hanegraff where he documents the danger of this deceptive brand of revivalism.
What I am referring to by the term charismatic Calvinist is defined by the term, continuationism or non-cessationism. Continuationists maintain that certain gifts of the Holy Spirit have continued to this present age. The opposite of continuationism is cessationism which maintains that gifts of the Holy Spirit like, prophecy, tongues/interpretation, healings, exorcisms, and miracles ceased after the apostolic era since they were needed only to authenticate apostolic authority (Grudem, 1031).
There are varying degrees of continuationism. Some well-known theologians/preachers who are considered to be continuationists are John Piper, Wayne Grudem, Mark Driscoll, Justin Taylor and C.J. Mahaney to name a few. Al Mohler was associated with Neo-Calvinism in the aforementioned Times article but I've not found any personal statement thereof so I am uncertain of this. I won't delve into all the specifics of continuationism. Rather, I'll provide some links to some studies and interviews that should shed some light on this idea.
One well-respected New Testament and theologian in the SBC wrote a paper that does a good job of defining some terms in this blog post. Check this out.
You should also read this interview with Wayne Grudem: Continuationism and Cessationism: An Interview with Dr. Wayne Grudem.
For a good study on "signs and wonders" I would strongly recommend this article by John Piper: Signs and Wonders: Then and Now as well as his fourteen-part sermon series on Are Signs and Wonders for Today?
This should at least get you started in understanding what is meant by the terms: Neo-Calvinism, Neo-Reformed and Charismatic Calvinism.
Finally, I'm not sure what I'm going to do with the domain name, CharismaticCalvinist.com, maybe Pastor Driscoll will offer to buy it from me.
Today we see our own slender resources, our poverty in fact; but this is undoubtedly the punishment we deserve, as the reward for our own ingratitude. For God’s riches are not exhausted, nor has his liberality grown less; but we are not worthy of his largess, or capable of receiving all that He generously gives.” (Calvin on 1 Corinthians 14:32; The First Epistle of Paul the Apostle to the Corinthians, p. 305).
Referenced:
Grudem, Wayne. Systematic Theology: An Introduction to Biblical Doctrine. Grand Rapids: Zondervan Publishing House, 2000.