Answer
If the equation has a higher exponent on the $x$ value than the $y$ value, then the equation is that of a parabola. (Parabolas have a power of $2$ on the $x$ exponent and a power of $1$ on the $y$ exponent.)
If the coefficients of $x^2$ and $y^2$ are different and are on the same side of the equation, then the equation is that of a hyperbola. (Hyperbolas usually have a positive $x^2$ and a negative $y^2$.)
If the coefficient on either $x$ or $y$ is not equal to $1$ while the other side of the equation is equal to 1, then the equation is that of a ellipse. (Ellipses have the equation $x^2/a^2+y^2/b^2=1$.)
If the equation is not a parabola, ellipse, or hyperbola, then the equation is of a circle.
Work Step by Step
Circle: $r^2=(x-h)^2+(y-k)^2$
Parabola: $y=a(x-h)^2+k$
ellipse: $x^2/a^2+y^2/b^2=1$
Hyperbola: $x^2/a^2 -y^2/b^2 =1$
Let's assume, for the set above, that $r=1$, all equations are centered at the origin, and that the x-intercepts for the ellipse and the hyperbola are $(-1,0)$ and $(1,0)$ and the y-intercepts are $(0,-1)$ and $(0,1)$. That would give us the following:
Circle: $1=(x)^2+(y)^2$
Parabola: $y=a(x)^2$
ellipse: $x^2+y^2=1$
Hyperbola: $x^2-y^2=1$
Circle: $1=(x)^2+(y)^2$
ellipse: $x^2+y^2=1$
Hyperbola: $x^2-y^2=1$
Circle: $1=(x)^2+(y)^2$
ellipse: $x^2+y^2=1$
Circle: $r^2=(x-h)^2+(y-k)^2$
ellipse: $x^2/a^2+y^2/b^2=1$
