In the middle of the Sermon on the Mount, Jesus explains the value of having "treasures in heaven," i.e. doing and concerning oneself with spiritual and eternal things, and then moves on to admonitions not to worry about earthly things. In tying these two topics together, Jesus makes the oft-repeated statement: "But seek first the kingdom of God and his righteousness, and all these things will be added to you" (Matthew 6:33).
What is the kingdom of God in the first place? It is an entity with both spiritual and physical manifestations, depending on the period of history, which indicates God's special rule through Jesus Christ. This kingdom was first established during Jesus' earthly life and continues still at a spiritual level made up of those who believe in Him for their salvation. After the tribulation, Jesus will reign for 1,000 years, a period called the Millenium, which fulfills promises made to Israel and Jesus for an earthly reign of Messiah – the physical kingdom of God.
With the meaning of the kingdom established, how does one "seek" it? When the context is taken into consideration, Matthew 6:33 appears to act as an intersection of the two previous topics, suggesting that if you concern yourself with spiritual things (seek the kingdom of God), the earthly concerns will "be added unto you" (essentially, provided for). To seek the kingdom of God is to intentionally focus on and fully experience relationship with God, usually through avenues He has provided such as prayer and meditating on His word. Seeking the kingdom of God does not stop there, however; it also includes sharing the fruit of relationship with God with others through things like evangelism, practical giving, and expressing love to others through whatever means God has specially gifted you or directed you.