The Greek word for baptize is "baptizo."  New Testament Greek, the same basic meaning has been retained: "To immerse, submerge, dip or plunge." Jesus is stating that the disciples will indeed be plunged into the same sufferings that He will experience.

There are many forms of persecution. Having your life threatened because of your faith in Jesus is one way you can be persecuted. 

History shows that the church has always flourished under persecution with increased numbers and zeal. During intense, life-threatening persecution, people's priorities get straightened out and the Lord assumes His rightful place. 

This always works for our good, regardless of what our outward circumstances might be. It helps to recognize that it is not you that they are persecuting, but rather Christ in you. You are actually taking part in His sufferings and will share in His rewards. 

With this in mind, we can actually shout and leap for joy in times of persecution.