Deuteronomy 6:4-5 says: "Hear, O Israel: The LORD our God, the LORD is one. You shall love the LORD your God with all your heart and with all your soul and with all your might." (ESV) This is called the Shema and orthodox Jews consider reciting it—morning and evening—to be among their most sacred duties. And when Jesus was asked which of the commandments was the greatest, this is the one He quoted, quickly adding that the command to "love your neighbor as yourself" was no less great (Leviticus 19:18).
The references to heart and soul and might (strength) are meant to convey the idea of the whole person. So the Lord is commanding that we love God with everything in us.
Having said that, we need to quickly acknowledge that this command can never be fully obeyed by sinful human beings. To love God perfectly, 24 hours/ 7 days a week without fail is outside of our ability. That's why Jesus needed to come and die on the cross for our sins. Everyday we fall short of this lofty ideal. And yet it ought to remain the goal of our hearts.
In 2 Kings we read of a king of Judah who loved the Lord as much as was humanly possibly. Of King Josiah it is said: Before him there was no king like him, who turned to the LORD with all his heart and with all his soul and with all his might, according to all the Law of Moses, nor did any like him arise after him. 2 Kings 23:25 (ESV)
When you read 2 Kings about the life of King Josiah you will see all the things he did and how his love for the Lord motivated him. Obviously the way you express your love for God is going to be different, but one thing will be the same: your love for God will consume you.
To love God with all your heart, soul, mind and strength is to live for God above everything else. It is to use all that God has given you—your heart, your thoughts and even your physical energy for Him and His purposes.
Essentially it's the opposite of how most of us live our lives, because if we're going to be honest we all spend the majority of our time living for ourselves. Our goals and efforts are all directed at being happy and feeling fulfilled. But that is a self-centered existence. God wants us to live our lives for Him—to please Him and serve Him. To do that with nothing standing in the way is to love the Lord with all your heart, soul, mind and strength.