A period is how long it takes for the wave to complete one full cycle.
Sin(x) has a period of 2 in radians, or in degrees. This means that if you draw the graph of sin(x), the graph would take 2 or to complete one full cycle. Which is a period.
Sin(2x) has a period of half of sin(x)
Sin(0.5x) has a period of double of sin(x)
The 2 in front of the x doubles the speed of one wave, which means that it halves the time, and since a period is the length of time for one full cycle, the period is halved as well. Vise versa for the 0.5.
Just think of it as this way: