Why don't planets have Circular orbits?
Because orbits are general conic sections. Why this is true is another fascinating question in and of itself, but for now I'll just assume it. The point is that circular orbits are special examples of general orbits. It's perfectly possible to get a circular orbit, but the relationship between the bodies' velocities and separation needs to be exactly right. In practice it rarely is, unless we plan it that way (e.g, for satellites).
If you threw a planet around the sun really hard its path would be bent by the sun's gravity, but it would still eventually fly off at a tangent. Throwing it really hard would make it almost go straight, since it moves by the sun so quickly. As you reduce the speed, the sun gets to bend it more and more, and so the tangent is flies off on gets angled more and more towards moving backwards. So general hyperbolas are possible orbits. If you move it at the right speed, then it'll be just slow enough that other tangent points 'exactly backwards', and here the motion will be a parabola. Less than this and the planet will be captured. It doesn't have enough energy at this point to escape at all.
A key realization here is that the path should change continuously with the initial speed. Imagine the whole path traced out by a planet with a high velocity. An almost-straight hyperbola, say. Now as you continuously lower the velocity, the hyperbola bends more and more (continuously) until it bends "all the way around" and becomes a parabola. After this point, you'll have captured orbits. But they have to be steady changes from the parabola. All captured orbits magically being circles (of what size anyway, since they have to start looking like parabolas at some point?) wouldn't make any sense. Instead you get ellipses that get shorter and shorter as you get slower. Keep doing this, and those ellipses will come to a circle at some critical speed.
So circular orbits are possible, they're just not general. In fact, I'd say the real question is why the orbits are often so close to circular, since there are so many other options!
This is because nature is not often as perfect as we tend to imagine it to be :)
In reality, when close to the Sun, the Earth has a little "too much speed" for it to stay that deep in the Sun's gravity well. In other words, the local spacetime curvature induced by the Sun is not strong enough to keep the Earth as close to it as it is, given that it also moves sideways; the Earth will start to "climb out" of the gravity well.
Building on your intuitions from everyday life, you know that if you move fast at the bottom of a hill, you'll be able to go up the hill, at the expense of your speed. You'll go higher and higher, but also slower and slower. The same holds in the context of the Earth. The further out of the gravity well it climbs, the slower it will move. At a certain point it will have climbed so far out that it actually moves too slow to stay as far out of the well as it is; it will start falling back in.
If you consider also the Earth's sideways motion and assume the Earth's motion is exactly perpendicular to the line of sight between the Earth and the Sun at the point where it moves slowest, it is not hard to imagine that the point where the Earth has its lowest speed is on the exact opposite side of the Sun as where it will have its highest speed. From that, it is not hard to imagine that the net result of all this will be an elliptic orbit, rather than a perfectly circular one.
Now, this does not mean it is not possible to have circular orbits. Of course, if the Earth would have had the exact amount of energy required to stay on the exact same distance from the Sun at all times, the orbit would have been circular. But this is nice in theory, but practically impossible to realize given that even the slightest perturbation from Jupiter or asymmetry in the Sun or whatever would cause the Earth's orbit to start deviating from a circle. Practically speaking, a circular orbit is a limit case that you can get arbitrarily close to, but never quite reach.
I think this video will help. Look at the motion of the balls in the cone; virtually the same principles apply for this case.
For completeness: there are many details I have omitted here that influence the exact shape of the orbit -- the "hill" analogy (or bowling-ball-on-a-rubber-sheet analogy, as it is often depicted in movies or documentaries) is an inaccurate and incomplete representation of the true nature of spacetime. Also, the other planets, remote galaxies, in-homogeneity of the Sun's gravity field, etc. distort the Earth's local spacetime. But all these effects are relatively small, and can safely be ignored to understand the ellipticity of the orbit.
Note: This answer was written for the original version of the question, in which the OP asked about spherical orbits (in which the plane of the orbit may change due to angular-momentum exchange), rather than circular/eccentric ones (in which the plane of the orbit stays constant in time). This answer is not relevant for the question as it currently stands, but I'll leave it here anyway, in case it's ever useful.
The technical reason is conservation of angular momentum; if the Earth were to move out of its orbital plane, that would change the direction (though not necessarily magnitude) of the angular momentum vector.
A less technical way of thinking about it is just with Newton's second law: the acceleration of an object is proportional to the force on that object. In particular, the acceleration and the force are in the same direction.
Let's say that the direction the Earth is traveling is "forward" and the Sun is on the "left". Then, to move in a spherical orbit rather than a circular or elliptic orbit, the Earth would have to move "up" or "down". But since we just defined "forward" as the direction it is currently going, that means Earth would have to accelerate "up" or "down". And Newton's second law tells us that would require a force in the up or down direction. The Sun's gravity doesn't give us a force in this direction; it only pulls to the "left". It's possible for the Sun to be pulling too hard or too soft to keep the Earth going in a circle, so Earth can go a little to the left or right (and in fact it does go left and right in an ellipse), but never up or down. So the Earth will continue gliding along in its orbital plane, rather than a more general spherical orbit.