Transistor as a switch
You want the transistor to allow 10 mA to flow thru it C to E.  From the red box, you can see that the transistor has a gain of 100 at 10 mA.  That means 10mA / 100 = 100 &micro;A is the minimum base current that just guarantees the transistor will conduct the current you want.

Since you are switching this current, not trying to control it proportionally, you should overdrive the transistor somewhat.  For switching like this, and considering the currents are low, I'd use a factor of two at least unless there are some other considerations you haven't mentioned.  That means you want to drive the base with at least 200 &micro;A when switching on.

The yellow box only tells you how much inescapable C-E voltage drop the transistor will have even when driven fully on.  For example, if you are switching the ground to something connected to a 12 V supply, then that load will actually only see 11.8 V across it, not the full 12 V.  In most cases that's good enough, but you have to consider this to make sure.  If you're driving a LED, for example, there will be 200 mV less across the series resistor relative to a perfect switch, which in that case is easy to compensate for by making the resistor a little smaller.

The middle section tells you what the B-E drop will be, which is 850 mV max for 1 mA base current.  Since it doesn't say what it is at your base current, use that as your worst case too.  It's not going to be higher at lower base current.

Let's say you want to drive the transistor from a 3.3 V logic output.  You will need a resistor between the output and the base, with the emitter tied to ground.  Since the base can be as high as 850 mV, that leaves 2.45 V across the resistor.  (2.45 V)/(200 &micro;A) = 12.25 k&Omega;, which is the maximum allowed base resistor to meet minimum base current we decided on earlier.  A little more base current won't hurt, so the common value of 10 k&Omega; will do nicely.

Depending on how fast the transistor needs to switch off, you may need to use a lower value or a C or R-C speedup circuit.  If you're just switching a indicator LED, just the 10 k&Omega; base resistor will be fine.