Why do we slip while trying to run on a wet floor but we don't slip when we walk slow and steady? When we try to walk( or run) on a wet surface we tend to slip but on the same wet surface when we walk slowly, it is likely that we won't slip ? 
Why is it so? What is the role of friction here ?
 A: 
While walking the static friction acts on your feet to hold it in a place and when you apply a force via pushing the ground backward it in turn pushes you forward (Newton's third law). Here you would see that your bones push your feet backward and hence are pushed forward and in a similar manner your whole upper body is pushed forward. This generates a torque in your body and the axis being the point of contact to the ground, held via frictional force. This makes you lean forward and then you put your second leg forward (so as to stabilize yourself).

Now when you are walking on a wet floor then coefficient of static friction is small and hence when you try to walk faster (which requires larger force to generate larger torque) via applying a larger force you easily get slipped because your applied force may exceed the static friction  but if you apply lesser force (which in effect would cause lesser torque and less speed) you can walk more easily and therefore don't slip. 
A: Water on a surface can lower the co-efficient of static friction, which will make it take less force to break static friction (shoe not slipping), and become kinetic friction (shoe slipping). Moving slowly on a wet surface usually causes less horizontal forces which can break  the static friction between the shoe and the floor. Hydroplaning can also be a problem when moving quickly, with hydroplaning you are actually held up by the water for the short time it takes for your weight to push the water out from between the shoe and the surface.
A: look at this diagram $F=F(t)$

where:
$F_i$ is your foot force 
$F_{\mu\,k}=\mu_k\,m\,g$ is the kinematic friction force between your foot and the surface.
As long as your foot force is less then the kinematic friction force you don't move
if you run then your foot force gradient is greater then as  if you walk, thus:
$\frac{dF_1}{dt} > \frac{dF_2}{dt}$
and 
$t_1 < t_2$
where $t_i$ is the time where your velocity is greater then zero  
I want to defined the slip according to  this equation: 
$s_L=\frac{v(t)-vs}{v_s}$ 
with :
$v(t)=\int\left(\frac{F(t)-F_{\mu\,k}}{m}\right)\,dt$  your velocity and 
$v_s$ is a arbitrary reference velocity.
thus:
the slip due to force $F_1(t)$ is :
$s_{L1}=\frac{v_1(t)-v_{s}}{v_{s}}$
and  due to force $F_2(t)$ is
$s_{L2}=\frac{v_2(t)-v_{s}}{v_{s}}$
thus : 
if $\frac{dF_1}{dt} \gg \frac{dF_2}{dt} \quad \Rightarrow s_{L1} \gg s_{L2}$ .
the slip when you run is much greater then the slip when you walk.
A: It is static friction that prevents your foot from slipping. But static friction has a limit:
$$f_s\leq \mu_sn$$
By taking a step, you exert a backwards force. Static friction then appears as the equal but opposite reaction force and pushes forward to avoid you slipping. As you walk faster, you exert more force. Since static friction must equal your force, it must be larger as well.
If the limit of static friction is below the force needed for walking fast but higher than the force needed to walk slowly, then you see the effect you describe: Then you can walk but not run without slipping.
On a slippery surface, the friction coefficient $\mu_s$ is greatly reduced and thus the limit is much smaller - maybe so small that even walking require too high a static friction that above its limit. A soaped bathtub or an icy sidewalk would be such examples. 
A: I don't have the ability to provide fancy animations or pictures but i think i can explain it theoretically
Slipping occurs when the torque applied by friction exceeds the anti-torque provided by the body to remain stable, when we run over a slippery surface in a fast manner(we exert more force on ground and vice versa), it results in a change in friction force by a large magnitude( due to decrease in friction coefficient) the body can't adjust the anti-torque resulting( if the person's reaction time is large) in us losing our balance and falling down but if we walk slowly the relative change in torque is minute so as to remain in a balanced state and not falling, I hope my reasoning is correct.
Thanks and regards
A: 
When we try to walk( or run) on a wet surface we tend to slip but on
  the same wet surface when we walk slowly, it is likely that we won't
  slip ? Why is it so? What is the role of friction here ?

When we walk or run our foot applies a pushing force backwards against the ground. Per Newton's third law, the ground applies an equal and opposite reaction force forward. See the free body diagram of a runner below. It applies as well to a walker.
The ground reaction force on the person is resolved into the static friction force parallel to the surface and the reaction force normal to the surface. It is the forward acting static friction force that propels us forwards and keeps us from slipping. The faster we walk or run the more we lean forward and the greater the magnitude of our backward pushing force on the ground, meaning the magnitude of the maximum static friction force needed to prevent slipping has to be greater.
The problem is if we push back too hard (walk or run too fast) on a wet or icy surface our backward force may exceed the maximum possible static friction force causing us to slip. The maximum static friction force is $F_{max}=μ_sN$ where $μ_s$ is the coefficient of static friction between our foot and the surface and $N$ is the normal force, or component of the walker/runner's pushing force shown in the diagram below. On a wet or icy surface $μ_s$ is very low, reducing the maximum static friction force and making it easier to slip.
When a person walks very slow, the angle $θ$ increases thereby reducing the static friction force needed to propel the person forward making it less likely that the maximum possible static friction force will be exceeded  and slipping occur.
Hope this helps.

