The cross was death in Roman culture. Jesus is telling His disciples to die to themselves in order to serve God fully. The image of "taking up your cross" is one of a condemned criminal carrying the cross bar of his own execution through the streets of the city and enduring the insults and humiliation of the crowds along the way.
Denying ourselves and taking up our cross means we have decided to let go of our selfish ambitions and own ways and fully dedicated our lives to following Jesus.
Jesus is hitting the disciples and hitting us right between the eyes with the call to seek Him first by "losing our lives" for His sake...it's by abandoning selfishness and handing our lives over to God that we find "life."
Our souls are more valuable than anything this life can offer and since God is the author of life He knows better than we do that living for Him (and not for ourselves) is what it means to be a disciple.