A good practice is to draw a diagram based on the description of the problem.  The question talks about a $120^\circ$ sector of a circle of radius $1$.  Can you draw that?  First draw a circle of radius $1$, and cut out a $120^\circ$ sector.

Then, the question wants to find out the largest circle that can be fit into the sector above.  Can you draw that?  To be the largest, this circle should touch the arc and the two straight sides of the sector.

Now it is time to set up equations.  But let's connect some lines first.  When two circles are tangent to each other (internally or externally), we can connect the centers because they line up with the tangent point.  The sector is part of a big circle of radius $1$.  Let the small circle have radius $r$.  So the distance between the two centers is $1-r$; also the line bisects the angle $120^\circ$.

Now, because the small circle is also tangent to the straight side of the sector, we can connect the center of the smaller circle and the tangent point, because that radius is perpendicular to the side.  So we have a right triangle whose hypotenuse is $1-r$, one leg is $r$, and the angle opposite $r$ is $60^\circ$.

Based on the side ratios of a 30-60-90 triangle, we have equation $$(1-r)\frac{\sqrt{3}}{2} = r,$$ and just solve for $r$.