Although most often called the "Kingdom of God," occasionally other terms are used in describing the Kingdom. Three of the writers of the Gospels—Mark, Luke and John—use the term "Kingdom of God" to refer to the Kingdom by name.
"Kingdom of heaven" is a term used exclusively by Matthew, with 32 references in his account of Jesus Christ. However, he uses the terms "kingdom of God" and "kingdom of heaven" interchangeably. In Matthew 19:23,24, he uses the terms in consecutive verses, clearly implying that they were synonymous. Often he calls it simply "the Kingdom."
Why did Matthew call it "the kingdom of heaven"? Because heaven is where God is, as Jesus Christ made plain (Matthew 5:34, 45, 48). Matthew makes it clear that the Kingdom was not, at that time, an earthly monarchy like the kingdoms around them. However, he understood that it was a kingdom to come, for which Christ's followers are to pray (Matthew 6:10).
The apostle Paul usually refers to it as "the kingdom of God." However, acknowledging the role of Jesus Christ as the Ruler of that Kingdom and the way by which we enter that Kingdom, he also calls it "the kingdom of Christ and God" (Ephesians 5:5). He also expresses the deep, loving relationship between God the Father and Jesus Christ by calling it "the kingdom of the Son of His love" (Colossians 1:13).
The apostle Peter, also acknowledging the centrality of Christ's role in the Kingdom, refers to it as "the everlasting kingdom of our Lord and Savior Jesus Christ" (2Peter 1:11). Jesus Christ is our Lord and Master now, and He will reign supreme in the coming Kingdom (Revelation 17:14; 19:16). As Savior of mankind, He is "the door" and "the way" by which we have access to God the Father and salvation in God's Kingdom (John 10:9; 14:6).