What do we mean when we say constant or uniform? What are the meanings of the terms constant and uniform in general?
Constant: is it something that doesn't change or changes constantly?
If yes, then what is uniform? Are they both same?
 A: These aren’t technical terms whose difference in meaning is taught to new students, and you could get pretty far by using them interchangeably.
One definition that feels natural to me might be that “constant” refers to quantities which don’t change with time, while “uniform” refers to quantities which don’t vary over space. That is, “constant $\phi$” means $d\phi/dt = 0$, while “uniform $\phi$” means $d\phi/dx = d\phi/dy = d\phi/dz = 0$.
For instance, imagine that you’re at a foundry and you’re watching a blacksmith use an oven to change the temperature of something that will soon be a sword.  One way to do this is to put the entire sword into the oven.  Initially the entire sword is at room temperature; later it is at the same temperature as the oven. If the tip and the blade and the hilt of the sword all heat up at the same rate, you might say that its temperature is uniform the entire time it’s in the oven; but it’s not constant, because it starts off cold and later it’s hot.
A more common manipulation is to put the point of the sword in the oven but leave the hilt outside.  You can imagine that, if you did this for long enough, the temperature might come to some steady-state equilibrium, where the temperature of the point is close to the temperature of the oven, the temperature of the hilt is close to room temperature, and the temperature of the blade is somewhere between. This equilibrium distribution is definitely non-uniform.  If you were to refer to such a temperature distribution as “constant” a physicist would probably get squirmy and try get you to use dynamic-equilibrium language, but if you clarified “I mean it’s constant in time” they might be mollified and let you finish making your point.
When talking about dynamics near Earth’s surface we are much more likely to talk about a “uniform gravitational field” leading to “constant-accelerated motion” than with the adjectives reversed.  In reality Earth’s gravitational field is not uniform: it gets weaker as you go higher up, and it points in different directions if you move far enough that you care about Earth’s curvature.  But when you go to find out how the velocity has evolved with time,
$$
\vec v_f - \vec v_i = \int_i^f dt\ \vec a,
$$
the integral is trivial if the acceleration $\vec a$ is independent of time.  In a free-fall system where the acceleration comes entirely from gravity, one way to accomplish this time-independent acceleration is to have the gravitational field unvarying as you move around.  That is to say, constant acceleration is a consequence of motion in a uniform gravitational field.
An introductory physics class spends lots of time talking about “uniform circular motion.” In uniform circular motion there is definitely something about the motion that is “always the same,” but a lot of ink is spilled about how the velocity can’t be constant, because its direction changes and there is a nonzero net force involved.  The solution is to construct a rotating coordinate system, so that the force appears along the center-pointing (centripetal) direction.
(I’m not actually convinced that “unvarying in a carefully-chosen coordinate system” is what a normal physicist means when she tells you to say “uniform circular motion” instead of “constant circular motion.” But the correspondence is satisfying enough — and that particular usage is common enough — that I’ll leave it in here.)
