Cassini ovals

(* If you view {x,y} as  r*{cos(t), sin(t)}, then CP is the "r^2". *)
CP[t_,a_,b_] =a^2*Cos[2*t] + Sqrt[b^4-a^4*Sin[2*t]^2] 
xP[t_,a_,b_]=Sqrt[CP[t,a,b]]*Cos[t]
yP[t_,a_,b_]=Sqrt[CP[t,a,b]]*Sin[t]

(* This is needed only when the ovals are to look like two eggs. It is not needed if the curve you want to plot is a single peanut-like (or melon-like) shape. *)
CM[t_,a_,b_] =a^2*Cos[2*t] - Sqrt[b^4-a^4*Sin[2*t]^2] 
xM[t_,a_,b_]=Sqrt[CM[t,a,b]]*Cos[t]
yM[t_,a_,b_]=Sqrt[CM[t,a,b]]*Sin[t]

COval[a_,b_,mycol_]:=
ParametricPlot[{{xP[t,a,b],yP[t,a,b]},{xM[t,a,b],yM[t,a,b]}},{t,0,2*Pi},PlotStyle->mycol]

Show[COval[1,0.9,Red],COval[1,1,Green],COval[1,1.1,Blue],PlotRange->All]

Show[COval[0.9,1,Red],COval[1,1,Green],COval[1.1,1,Blue]]

Further reading/browsing:

You may find out more about properties of Cassini Ovals following these clickable links:
 -  Wikipedia, 
 -  Wolfram MathWorld, 
 -  webpages by Paul Bourke and by Ryan Byrd (University of Georgia).
Some applications, including that in radiolocation, are briefly described in a paper
by Mümtaz KARATAŞ from the Naval Postgraduate School.

Heart curve

xH[t_]=16*Sin[t]^3
yH[t_]=14*Cos[t]-5*Cos[2t]-2*Cos[3t]-1*Cos[4t]
ParametricPlot[{xH[t],yH[t]},{t,0,2*Pi}]

The specific coefficients above may be changed; this will, of course, change the curve.
More on similar curves can be found in:
 -  Wolfram MathWorld; 
 -  Mathematische Basteleien (despite the main site being in German, this article is in English; it contains more entertaining information than the MathWorld article).

Petal curves

rpc[a_,b_,k_,t_]=(a+b*Cos[k*t])/(a+b);
   (* The denominator of the above expression makes it have the maximum value of 1.  *)
xpc[a_,b_,k_,t_]=rpc[a,b,k,t]*Cos[t];
ypc[a_,b_,k_,t_]=rpc[a,b,k,t]*Sin[t];
aa=3; bb=2; kk=3; ParametricPlot[{xpc[aa,bb,kk,t],ypc[aa,bb,kk,t]},{t,0,2Pi}]