The meaning of the cross to the Christian is exactly what it meant for our Lord Jesus: suffering, persecution, oppression, opposition when trying to do our Father's will. Our Lord always obeyed His Father's will under unfavorable conditions, and that definitely meant for Him wearing the cross. The same goes for the real Christians. The cross to them means opposition when they seek to serve His cause and let the light of the truth shine, just as was the case with our Lord Jesus. The following Bible verses address cross-bearing:

Matthew 16:24, “If any man will come after Me, let him deny himself, and take up his cross, and follow Me.” Cross-bearing is closely related to self-denial, and yet a distinction between them may be noted. Self-denial relates more particularly to passive obedience and endurance for the Lord's sake; cross-bearing has to do more especially with activities in the Lord's service, which we find to be contrary to our natural inclinations…Our cross-bearing, however, may be seen to some extent at least, by those who are in close contact with us, and especially by those who are walking in the same “narrow way.” 

Luke 14:27, “Whosoever doth not bear his cross, and come after Me, cannot be My disciple.” The Lord’s cross-bearing brought upon Him the envy, hatred, malice, strife, persecution, etc., of those who thought themselves to be God's people, but whom our Lord, who read their hearts, declared to be of their father, the devil. Since we are walking in the same “narrow way” that our Master walked, we may reasonably expect that our crosses will be of similar kind to His–oppositions to our doing the will of our Father in heaven–oppositions to our serving His cause and letting the light shine out as our Master and Leader directed.

1 Peter 4:12-13:"Beloved, think it not strange concerning the fiery trial which is to try you, as though some strange thing happened unto you: But rejoice, inasmuch as ye are partakers of Christ’s sufferings; that, when his glory shall be revealed, ye may be glad also with exceeding joy."

2 Timothy 3:12 "In fact, everyone who wants to live a godly life in Christ Jesus will be persecuted."