Can someone explain the equation for entropy? Here's the equation:

$\Delta S = \frac{\Delta q}{T}$

The confusion I have with this equation lies in the variable for temperature, $T$.  Why is the temperature constant in this equation? Why can't the equation include $\Delta T$ instead of just $T$ and, surely, if $\Delta q$ is negative or positive the temperature must decrease or increase, respectively.
Your relation is not correct, so you must not understand something. If I were you I will asked the question in Physics SE but nevermind. 

The good relation is : $$\mathrm{d}S=\frac{\delta Q}{T}$$ 

I let Richard Feynman explain it to you here (when you click on the link wait few seconds)

I hope it will help you, have fun, live long and do chemistry !