What happens to our force when we walk on ice? I'm scratching my head a lot in trying to understand friction.
So far I understand that "without friction we would not be able to walk". But that sounds really vague and unclear, so much in fact that it doesn't make any sense to me unless I give it one more thought.
As far as I understood: the force of friction appears when there are two contact surfaces that interact against each other (I guess the same thing as "they are in relative motion to each another"). As long as the threshold of this friction is not surpassed the Force of friction will adjust itself to any existing opposite force, balancing it.
If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground that same friction will apply the equal force to my foot, allowing me to move forwards.
For this to happen, the force of friction must adjust itself to my contact force? What force does my foot exert? I'm confused...
If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip. But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.
There is something I can't understand from friction...
 A: 
the force of friction appears when there are two contact surfaces that interact against each other

If you by "interact against" mean "try to slide over", then correct.

As long as the threshold of this friction is not surpassed the Force of friction will adjust itself to any existing opposite force, balancing it.

Indeed. And to be accurate, every mention of friction here refers to static friction (as opposed to kinetic friction), which is what happens when sliding is prevented (when something tries to slide but doesn't).

If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground that same friction will apply the equal force to my foot, allowing me to move forwards.

Yes, although a bit combersome sentence. A surface does not "have friction". Rather, it has a roughness so that friction can appear when sliding against another surface wants to begin.

For this to happen, the force of friction must adjust itself to my contact force? What force does my foot exert? I'm confused...

When you apply a backwards force with your foot - we can call it a stepping force, if we will - then as per Newton's 3rd law the ground responds with an equal but opposite forwards static friction. I call it a stepping force but there is no typical official name for it as a whole, as far as I'm aware. Depending on scenario you might also call it thrust or traction or the like as mentioned in a comment.

If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip.

What you mean here is that there is an upper maximum limit to static friction. But note, your stepping force cannot be bigger than this limit. Your stepping force can only exist if an equal but opposite static friction force also exists (again, this is Newton's 3rd law). If the ground cannot respond with an equal static friction force then it let's go, meaning the static friction force disappears.
Then you don't have to apply any larger stepping force since your foot is just slipping and sliding and not feeling any resistance to apply force against.

But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.

As described, it doesn't "go" anywhere because you can't apply a force against nothing. When you punch into empty air then you aren't applying any force; likewise when you step backwards you can only apply a force equal to whichever resistance your feet meet.
Instead your foot just slips and slides backwards. Kinetic friction would takes over now, and then your stepping force equals that one instead. But if no other force take over - if there is no other resistance against your foot now - then you will just be "running on the spot" and will never move. As if running on ice, or if you are running while hanging in free space in a space station.
This can in no way accelerate you forwards. Only a forwards-pointing force could do that. This is the reason that it is not your step which causes you to walk forward, it is indeed the friction from the ground.
A: What needs to be incorporated into this picture, is that a walking human in this context cannot be approximated by a point-like object - the approximation that we commonly make in introductory mechanics. Walking implies forces applied between different parts of the body, notable between the legs and the torso, as well as the forces acting between the feet and the ground, and the gravity.
Thus, the force that makes us move forward is not the friction force, but the force exerted by our muscles, making the parts of our body move in respect to our foot, held in place by a friction force. The foot meanwhile remains static - they do not accelerate.
The ankle exerts a force on the foot, and the foot exerts a force on the ground, which is counteracted by the friction force. If the static friction force is not able to hold the foot in place, as it happens when walking on ice, the foot will slip and accelerated by the ankle force.
A: I will try to explain some of this but will need to dissect your statements.

If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground that same friction will apply the equal force to my foot, allowing me to move forwards.

To be clear, the road does not have friction.  Friction exists between the road and your foot.  It is a property of both materials that are in contact.  If you look up coefficients of friction in a book you will not find the friction of concrete, but you will find different numbers for concrete-wood, concrete-concrete, concrete-rubber, etc.

For this to happen, the force of friction must adjust itself to my contact force? What force does my foot exert? I'm confused...

This is a common problem conceptually, especially when you consider Newton's 3rd law and the fact that the Earth is not an infinite ideal constraint.  You are in control of your foot and can choose to slide it along the surface of any other object.  There are two types of friction and our understanding of them is empirical, though we have equations describing them.  Static friction is what allows you to walk and this force will balance the component of force you exert on the ground that is tangent to the ground, as long as there is no relative acceleration.  In contrast, pushing down does nothing w/r to activating friction.  There is a limit to how much the static friction force can grow and that is equal to $\mu_s ||N||$, where $||N||$ is the magnitude of the normal force, the force perpendicular to the interface between the two touching objects.  In the example of you walking on the ground this would be due to the earth pushing you up, and by the 3rd Law you pushing the earth down.  Like friction, the normal force is understood empirically.  The force of gravity between you and the earth draws you and the earth together.  Once you and the earth touch it is the normal force that prevents you from merging with or passing through the earth.  And it is basically due to the strength of the molecular bonds in each solid.  In contrast, if you and the earth were in a liquid or gaseous state you would pass through each other and mix into something.  Going back to friction, once the sideways force you exert on the ground grows beyond the limit static friction is broken and slipping occurs.  The two touching bodies still have a degree of roughness and there is a force called kinetic friction that takes over.  Like the normal force both types of friction can be understood at the atomic level to be the result of the strength of molecular bonding between atoms in the surface, as well as the variations in height of each surface at a micro level.

If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip. But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.

First of all contact is needed for you to exert force on the earth so once you've "slipped" at that moment the force is likely gone.  If you maintain contact then the static force is replaced by the kinetic force of friction and the force you are exerting may have changed but is still there.  I think the part you are trying to get your head wrapped around is that you feel like you need the ground to push back to exert a force in the first place and that is true.  If you were standing on a frictionless surface (like ice) the you could not exert a sideways force on the ground.  You could however move you foot backwards relative to your body.  This would cause the elements of your body to redistribute in order to keep your center of mass fixed.  This is a requirement of nature, a conservation law.  As you throw your foot back hoping to grip the floor your torso moves forward.  Pieces of you accelerate but in opposite directions.  In the case of having a grip then loosing it because you pushed too hard what has happened is that you have changed the surface, and the relationship between you and the earth, to such a degree that it is now a different problem.  Slipping absolutely could mean accelerating depending on the state of the body.  To understand this you need to analyze the states in detail for a given problem.  When a ball rolls on the ground it maintains static contact with the ground at the point that is instantaneously touching it.  Thus friction makes it roll.  This is sometimes called rolling friction in text books.  Under these circumstances the ball will roll horizontally at a constant speed.  If at some point slipping occurs then that means the point of contact is lost and there is relative motion between the contact surface.  This will in fact create a horizontal deceleration (slowing down) of the center of mass of the ball.  Similar statements could be made about your foot.
A: *

*While walking there is another force acting which is force due to gravity.

*If there is not enough friction your foot will keep traveling backwards instead of staying put and gravitational force will pull the body down (slipping).

A: Your leg muscles apply a force to your foot, relative to the rest of your body. If your are on normal ground, the ground applies a force through friction to your foot. As a result, there is not net force, and your foot acts as an intermediary that allows your body to apply a force to the ground, and as a result there is a reaction force in which the ground applies a force to your body. Your foot remains stationary while the rest of your body moves forward. Walking consists of alternating which foot remains stationary, so that overall each foot moves with the rest of your body.
If you were to apply the same force to your foot while standing on a frictionless surface[1], there would be no (horizontal) force external to your body, so your center of mass would remain stationary. This would mean that your foot would move backwards, and the rest of your body would move forward slightly.

If I understand when walking, the road will have some friction and if I exert a backwards force against the friction of the ground

You are not exerting a force on the ground directly. You are exerting a force on your foot, the ground is exerting a force on your foot, and the force of the ground on your foot is accompanied by a force that your foot exerts on the ground.

But where does my force go? It makes me accelerate?

With no external forces, your body as a whole cannot accelerate. You can only have acceleration of one part of your body relative to the rest. Without the force of the ground to counteract the force of your muscles, your foot accelerates backwards, but the rest of your body has a forward force that matches the backwards force on your foot.

I don't think "slipping" means accelerate.

Slipping means moving past. Normally when you walk, at each moment there is at least one foot on the ground that is stationary. If you're slipping, that means that the foot on the ground is moving. Moving is not itself acceleration, but it is the result of acceleration.
[1]As others have pointed out, friction is really a property of the two substances in contact, not either individually, so a more precise phrasing would be "a surface that does not have any friction between in and your foot", but it's simpler to say "frictionless surface".
A: When we walk or run we apply a pushing force against the ground. The ground applies an equal and opposite reaction force on us.
. During walking or running the normal reaction force is greater than the gravitational force on the person in order to lift the person off the ground.
The static friction force that the ground applies to the person helps forward motion, and is equal and opposite to the parallel force the person applies to the ground. If there were no static friction force the person will slip.
If my "force" is bigger than the frictional force then the friction won't be able to provide that force in the opposite direction, so my foot will "continue its path" i.e. slip. But where does my force go? It makes me accelerate? I don't think "slipping" means accelerate.
If friction not sufficient enough then changes from static to kinetic friction.this value is less than the force you apply I.e the muscular force so though there is an opposition the opposition is just not enough to make you stop. In a way you will be accelerating
$a$=$(F-f_r)$/m
In short your force is causing your acceleration
A: You are floating free in two dimensions, just like an astronaut can float in three.
It is like floating weightlessly in space but limited to two dimensions.
On perfect ice, you are fixed in the top-down dimension, as gravity accelerates you to the ground, perfectly normal.
In the other dimensions, there is just nothing you can interact with in terms of force, analogous to floating free in a space station.
But limited precisely in the vertical direction, if the ice is perfectly horizontal. Your weight causes a force to the ground, and the ice gives you an equal force. It cancels out, you do not move in the vertical direction, and it will stay that way.
If the ice would not be perfectly orthogonal to the gravitation vector, locally, you would begin to slide on the ice (and could not avoid it).
