There are parallel passages in Matthew and Luke.
What Mark and the other Evangelists describe here are three refutations that Jesus made for the (unspoken) claim by some that By the prince of demons he is driving out demons (Mark 3:22).
First, he presents the common sense argument that if He were casting out demons, he would be working against Satan and not with him (Mark 3:23-26):
How can Satan drive out Satan? If a kingdom is divided against itself, that kingdom cannot stand. If a house is divided against itself, that house cannot stand. And if Satan opposes himself and is divided, he cannot stand; his end has come.
Second - and this is recounted in Matthew and Luke, but not Mark - he questions them regarding His disciples:
Matthew 12:27 (NIV)
And if I drive out demons by Beelzebul, by whom do your people drive them out? So then, they will be your judges.
Now if I drive out demons by Beelzebul, by whom do your followers drive them out? So then, they will be your judges.
The phrase that the NIV translates as your people and your followers in Matthew and Luke, respectively, is in both cases υἱοὶ ὑμῶν (uioi umōn), which literally means your sons. The understanding in antiquity was that Jesus was speaking here of the Apostles.1 (The ESV and other translations render your sons in both cases; I don't think there is a solid basis for the NIV's your followers in Luke). He referred to the Apostles as your sons to show the Pharisees that the path the Apostles had taken was open to them as well.
The Apostles had also been casting out demons, but they are not being accused. Jesus raises the point that since the Apostles are casting out demons by His (Jesus') authority, and the Pharisees had not accused them, then how could they also not be guilty?
Mark 3:27 (also Matthew 12:29 and Luke 11:21-22) contains the third refutation:
In fact, no one can enter a strong man’s house without first tying him up. Then he can plunder the strong man’s house.
The meaning here is that not only could Jesus not logically be casting out demons in the name of the devil, but that He would, in fact, have to have overcome the devil first to be expelling demons. "For that Satan cannot possibly cast out Satan", wrote John Chrysostom, "is evident from what has been said; but there is also no way to cast him out without first getting the better of him"2
1. See, e.g., John Chrysostom (d. 407), Homily XLI on Matthew