What is this diamond-shaped voltage source symbol, and how can I solve this puzzle?
The thing in the middle is a dependent voltage source. It is a voltage source that has a variable voltage, the value of which depends on the given formula. 

Now on to solving this puzzle... (it really is much more of a puzzle than a circuit)

The 6 V independent source forces the voltage between the two nodes to be 6 V. 

This means that the voltage across the dependent source must also be 6 V. As the +/- signs are reversed, this means that \$\alpha i_\Delta = -6\text{ V}\$. But we know that \$i_\Delta = -15\text{ mA}\$ looking at the current through the current source. 

Plugging in \$i_\Delta\$ into the first equation, we get

$$\alpha\cdot(-15\text{ mA}) = -6\text{ V}$$

So 

$$\alpha = 400\text{ Ω}$$

You can actually calculate the power of the current source without doing any of this, since \$P = IV\$ and you know both \$I\$ and \$V\$. Just multiply 6 V times 15 mA and you get 90 mW.