Ok, let's look at an example. We'll consider the differential
equation
y' = c + y^{2}.
Equilibrium solutions are when
y' = 0, or
c +
y^{2} = 0. This says that
y = +/sqrt(
c). What does this look like? If we
just plot it in Mathematica, we see

Plot[{Sqrt[c], Sqrt[c]}, {c,3,3}];
So it looks like if we look at a value of
c <0 we have
two equilibrium solutions, while for values larger than 0 there aren't
any. This means that we have to worry about two general regions: when
c<0, and when
c>0. These are the different values
of
c that we want to consider in our analysis, because they are
the regions in which things are different.
Before actually trying to do a bifurcation diagram, let's think about
what our state lines tell us. If c>0, we can see that
y' = c + y^{2} > 0, because the righthand side
is the sum of two positive numbers. Thus in this case the state line
(with no equilibria) is pretty easy to draw:
If
c < 0, then
y' = 
c +
y^{2}, so if
y^{2} >
c we have
y' > 0, and the solution has positive slope (is moving
upwards). Similarly, if
y^{2} <
c the solution has
negative slope and is moving down, as shown in the second state line
above.
Note that we could draw the second state line by choosing a value for
c and then seeing what happens  for example, if
c = 4, then we have y' = 4 + y^{2}. The righthand side of this is

Plot[4+y^2, {y,3,3}];
so for
y < 2 the derivative is positive, for 2 <
y < 2 it is negative and for
y > 2 it's
positive again  as shown in the state line above.
Finally, the bifurcation diagram is the plot of the equilibrium
solutions that we have at the top of the page, with solid and dashed
lines indicating attracting and repelling equilibrium solutions. As
shown in the state lines, the bottom one is an attractor and the top a
repeller:

Plot[{Sqrt[c], Sqrt[c]}, {y,3,3},
PlotStyle>{Dashing[{0.01,0.02}], GrayLevel[0]}];