Long story short, the equation you are looking for is this:
$${\Delta t'={\frac {\Delta t}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$$
Where $t$ is time, $v$ is a relative velocity and $c$ is the good old speed of light in a vacuum.
Explanation: supposing you and a friend sync clocks while both of you are at rest in relation to each other. You then change speeds relative to each other. If you are able to peek at their clocks from a distance, then, whenever you have measured a span of time $\Delta t$ on your clock, you will notice a span of time $\Delta t'$ has passed on theirs.
So for example, if you spend one hour moving at half the speed of light relative to your friend...
$${\Delta t'= {\frac {1h}{\sqrt {1-{\frac {(\frac {c}{2})^2}{c^{2}}}}}}} = {\frac {1h}{\sqrt {1-{\frac {\frac {c^2}{4}}{c^{2}}}}}} = {\frac {1h}{\sqrt {1-{\frac {c^2}{4c^{2}}}}}} = {\frac {1h}{\sqrt {1 - \frac{1}{4}}}} = {\frac {1h}{\sqrt {\frac{3}{4}}}} \approx \frac {1h}{0.866} \approx 1.154h \approx 1h 10m $$
That is, for every one hour that passes for you, it seems like approximately one hour and ten minutes have passed for your friend. Notice that this is reciprocal. Your friend will also notice that time is passing differently for you, just the same. When one hour has passed for him, he will see your own clock marking a span of approximately one hour and ten minutes.
The lengthy, parlance heavy explanation can be found in Wikipedia.