How do I connect a vibration motor to Arduino?
Lets break down the circuit and have a look why everything in that circuit is there and what it does.

First we have the 33R resistor before the motor which limits the current to the motor. This particular motor is rated at 75mA and has a starting of 85mA. The 33R resistor limits the current to 130mA maximum through the motor. If the risk of the motor pulling too much current is an issue then stick a fuse in-line

Then there's the motor, not much to say here really. It's whatever motor you're using.

We then have both a diode and a capacitor across the motor. I find this a bit strange, I've seen the use of flyback diodes, I've seen the use of snubbers, but both? The purpose of the diode is to allow a return path for any spikes caused by switching off power to the inductive load (the motor) which stops them from damaging the transistor. For the capacitor, just remove it, it's not got any use here really.

Then comes the transistor, it might be easier to think of a transistor as a switch (especially in this example). When you put a little bit of current on the base, the transistor 'opens' allowing the motor to connect to ground. The reason we have to use a transistor is because the outputs of an arduino are only able to supply a maximum of 40mA, we need more than that to run our motor. Fortunately, the transistor is not going to need any more than 5mA on it's base which is why we use it to 'switch in' components that require more voltage/current. The 1K resistor on the base is to limit the current coming from the arduino and preventing it from killing our poor transistor.

tl;dr - You need everything except the capacitor