What force enables us to walk? Friction or normal reaction? I know that this question has been asked here before, but I am a little bit confused with all the answers. So when we move, we apply force on the ground in the backward direction. So, is the ground applying a force on us in the forward direction by virtue of friction or by virtue of normal reaction? 
Some answers seem to suggest that it is the former, while some say that friction is the reaction force. I used to think that normal reaction is the force that only exists so that solid objects don't pass through each other.  
 A: Both are required for walking$^*$. 
You need friction to accelerate when you want to start walking, stop walking, change speeds while walking, etc. This is because you need a horizonal force to change your horizontal speed. This force is friction. It arises due to interactions between your feet and the ground you walk on. Therefore, by Newton's third law, the ground is pushed on by friction in the opposite direction of your horizonal acceleration.
However, don't discount the normal force. It is a vertical force (on level ground). Therefore this force is what keeps you from accelerating downward into the ground due to gravity. It also is one of the factors in determining how strong the previously mentioned friction force can be. A larger normal force typically means a larger possible friction force before sliding between your feet and the ground occurs. Therefore, without the normal force you wouldn't be able to walk either.

$^*$ Of course other forces like gravity, internal forces in your body, etc. are also important for walking. The physics of walking can get pretty complex. However you just asked about these two forces (friction and normal force), so I will just focus on those two.
A: Ice lacks the friction.  Water lacks the normal force.  Since walking on ice is difficult and walking on water is impossible (for most of us) I'd say they're both important.
A: Friction force is proportional to Normal force as well as perpendicular to it. (http://hyperphysics.phy-astr.gsu.edu/hbase/frict3.html). So both are related in my opinion. Normal force is not necessarily in the same line as gravity (for example when the surface is inclined).
It is the Friction force, ultimately that makes you walk. If friction coefficient is zero (for example on a glass surface if you apply a lot of oil, there will be close to 0 frictional coefficient) then you will not be able to walk because friction force(=friction coefficient * Normal force) will be 0. 
I might be wrong but this is my simple understanding.
A: You're able to walk because of friction, not the normal force. To see this, try walking on a low-friction surface such as wet marble. You will have difficulty, even though the normal force hasn't changed (it's still equal to your weight).
Another way of seeing this is to note that the friction's direction is horizontal while the normal force is vertical, and you're trying to move horizontally.
A: The existing answers are arguably incomplete. In walking, some friction involved is due to normal reaction.
There are two main kinds of friction. The first is that if both surfaces in contact have an uneven topology, then clearly when they are pushed together the bits that jut out will bang against each other and cause friction on the macro scale. The second is intermolecular attractions between the surfaces in contact, which is the main reason (real) rubber is good for making shoe soles non-slip.
It should be clear that the first kind is explicitly an instance of normal reaction forces underlying the resulting friction. But for the second kind, when a particle A on one surface is attracted to a particle B on the other surface, pulling A in one direction will result in a pull on B in a similar direction, and it simply does not fit the category of "normal reaction", even though it is the same underlying principle that forces must balance in equilibrium (which in this case corresponds to non-slipping).
A: Technically, both normal and frictional forces are required: 
The normal force supports you vertically by opposing the force of gravity pulling you down. The normal force also affects the maximum static frictional force, which is the smallest force required for your feet to slide, or the maximum amount of force you can apply horizontally to the ground without slipping. 
The frictional force that directly helps you to walk; it provides a force that “pushes” you forward, so directly it helps you to walk. 
However, the frictional force is ‘powered’ by the normal force (they are directly proportional so if there is no normal force, there is no frictional force) so if the normal force was not present, the frictional force would not either. 
