What is the cross in the mantra "carry your own cross"? Why did Jesus carry the cross and why did he say that everyone should take up their own cross?
And what is the cross anyway?
I believe that we can get a definitive answer on this from the Apostle Paul. He spoke of the Cross as the summation of Christ and His Mission to redeem man.

He stated that the Message of the Cross is the Power and Wisdom of God to those who are saved (1 Cor 1:18, 23), and emphatically stated that he would not preach or know anything other than "Christ and Him Crucified" (1 Cor 2:2). He taught in many places that the very life we have as Christians stems from being planted together with Christ in His Death (Gal 2:20, Romans 6:6). If we compare what Paul says about the Wisdom of God from 1 Cor 1:23 and Eph 3:9-10, we see that he is wrapping up in the term "Cross" the entire, eternal mission of Christ. We can confirm that with Phil 2:8-9. So, when we are to take up His Cross, the Cross, our Cross, being that we are planted together with Him in the likeness of His Death, it means to take up His Redemptive Mission. We are to "seek first the Kingdom of God and His Righteousness" (Matt 6:33) (also flows from Calvary [Rom 3-4]). We are to live by faith in His Blood (Rom 3:25). We are to look to Calvary.

Since Grace flows through faith (Eph 2:8-9), and Satan knows that in order for him to remove us from the Grace of God, he must destroy our faith, he knows that he must eliminate our faith in the Atoning Work of Christ that reconciles us to God. Therefore, with believing properly comes a struggle of faith... a good fight of the faith. (1 Tim 6:12). This struggle (fight, literally "agony") to keep one's faith off of self (the world, the flesh, and the devil all fight against this faith) and unto Christ is what it means to "take up one's Cross". It means to carry the Cross in your faith, which will allow Grace (Eph 2:8-9), which will allow the Holy Spirit to conform us into the Image of Christ  (Rom 8:29, 2 Cor 3:18), which will enable us to build the Kingdom of God in Resurrection Power (Acts 2:8; Romans 6:5).