The ParametricPlot command is pretty self-explanatory.  You provide x(t) and y(t) and a range on t, and Mathematica plots the curve.  The functions x(t) and y(t) are entered as a list {x(t),y(t)}.

ParametricPlot[{Sin[2t], Cos[t]}, {t,0,2Pi}]

The dynamic code below shows how a rolling circle traces out a cycloid.

max=2Pi;
x[t_] := t+Cos[3Pi/2-t]; y[t_] := 1+Sin[3Pi/2-t]; pp[u_]:=ParametricPlot[{{x[t],y[t]},{u+Sin[3Pi/2-t],1+Cos[3Pi/2-t]}},{t,0,max+1},Epilog->{PointSize[Large],Point[{x[u],y[u]}]}, PlotRange->{{-1, 2Pi+.5}, {-1/2,2.5}}]
Manipulate[pp[u],{u,0,max}]

The code below produces the graph of a cycloid.

r=1;
cycloid=ParametricPlot[{r t + r Cos[3Pi/2-t],r + Sin[3Pi/2-t]}, {t,0,4Pi}]

r=1;
cycloid=ParametricPlot[{r (t-Sin[t]),r (1-Cos[t])}, {t,0,4Pi}]

Clear[r]

x[t_] := r t + r Cos[3Pi/2-t]; y[t_] := r + Sin[3Pi/2-t]

Sqrt[x'[t]^2 + y'[t]^2]

r=1;
Integrate[Sqrt[x'[t]^2 + y'[t]^2], {t,0,2Pi}]

r=1;
Integrate[y[t] Sqrt[x'[t]^2 + y'[t]^2], {t,0,2Pi}]

The PolarPlot command is a bit more mysterious.  This command plots r =f(t), so we assume that r is a function of t.  We do not assume that r is positive.

PolarPlot[Cos[t],  {t,0, 2Pi}]