Most of the Christians I've spoken to use both. The meaning of the word testament is "promise". Some Christians believe that since God has given a new law/promise to the people through His son, Jesus Christ, then we should disregard the old ways and follow the new laws. Many Christians may dispute whether that should be done or not. The Jews are the ones that follow the Old Testament, also known as the Torah.
The four Gospels which make up the New testament were written by Mathew, Mark, Luke and John, but im not sure as to whether they were told to write the new tesament. Beacause The New Tesament talks about the Many things that Jesus Christ did while on earth right up until his death, but from what i can remember Mathew, Mark Luke and John wrote with help from the Holy Spirit.
Although the Hebrew Bible composed of the oral Torah as well as the written, the Old Testament is also part of the Torah. Basically, Old Testament is part of Judaisms core, but not the entirity of it.
The New Testament was a compiliation, I believe? The Gospels were written years after Christ had died, and then years later the Churches solidified the Bible which they thought composed accurate descriptions of Christ and geniune articles from the Gospels etc.
Christians usually don't take the Old Testament as seriously...If you've witnessed any Christianity-bashing, mostly anti-Christianity people just recite words and phrases from the Old Testament as an attempt to bash the Christian religion. Of course this is dismissed because Christians are mostly folllowers of the New testament.