When the Mythbusters tested it out, a pool of water stopped a 50. cal sniper rifle under 3 feet. Other weapons tend to go through water more easily, but high-velocity bullets just explode.
Why is that? Why can water stop most bullets, and why it can't stop certain types that effectively?
There are a couple ways to consider this situation.
A somewhat simpler explanation that doesn't account for everything would be the drag equation: $$ F_{D}\,=\,{\tfrac 12}\,\rho \,u^{2}\,C_{D}\,A$$
where $F_D$ is drag force, $\rho$ is fluid density, $u$ is relative velocity, $C_D$ is the drag coefficient for the specific shape and speed, and $A$ is projected area in the direction of travel.
Air is almost 800 times less dense than water. This means that going the same speed through air and water, a bullet will experience approximately 800 times more force in the water.
Because of the $u^2$ term, increasing velocity increases the force dramatically, which is why faster bullets are more likely to break.
Another factor is the shape, which will change the value of $C_D$. A more aerodynamic bullet would be less likely to explode on impact (though speed is likely to be a bigger factor).
A more complicated explanation (and probably just as important or more important) is the compressibility of water. Water is not very compressible, and when you strike it at very high speeds, it may not initially behave as fluidly as you may like/expect. It could act more like a solid surface as the bullet impacts. To analyze this would be somewhat complex and would involve analyzing transient effects and compressibility.
This is a bit more out of my wheelhouse, but something like the water hammer equation may be relevant (especially if you just flip the way you consider it, with the object approaching the water instead). This equation accounts for the equivalent bulk modulus, which describes the compressibility of the fluid. The sudden impact of fluids can be quite forceful.
See also Why is jumping into water from high altitude fatal? and related links.
I think that at the beginning the force is connected partly to the speed of sound in water$$ F\,\approx\,\,\rho \,u\,c\,\alpha\,S$$
where $\rho$ is fluid density, $u$ is relative velocity, $c$ is the speed of waves in water (almost 4.3 times faster then in air) and $\alpha$ is the coefficient for the specific shape and speed, and $S$ is projected area in the direction of travel. So, the estimation results in $10^5$N... $10^6$N during ~$0.1$ms.
Water molecules have hydrogen bonds in between them that hold them together, which means that they are not easily displaced. So when something collides with water or tries to pass through it, the water molecules resist being displaced and this creates a reaction force against the object. This is also what causes drag in general. The attraction force between the molecules determine its viscosity, and this in turn determines the resistance to displacement and therefore the drag. And water is much more viscous than air, so bullets slow down much more in water
As to your second question about why some bullets travel faster than others in water, its because the drag depends on the shape of the bullet. If a bullet has a smaller cross-sectional area and surface area, and is more 'pointy' towards the head, it will be more penetrative in water
