Recently I was taking a look at a video explaining the existence of fourth spatial dimension and thereupon that infinitely many spatial dimensions are possible. Also it showed that what Einstein told about time being a dimension itself was also incorrect as time is always present, even in one dimension. So how's it all possible, is the statement by Einstein not correct?
I think we'd need to check what is in the video to give a definitive answer. Do you have a link for it?
In relativity the number of dimensions is related to how we specify the position of a spacetime point. If I want to specify my position at the moment I started typing this I have to say where in space I am but also what time I started typing, so I'd specify it as $(t, x, y, z)$, where $t$ is the time I started typing and $x$, $y$ and $z$ give my position in space relative to whatever reference point is convenient. Specifying a point in spacetime takes four numbers, $t$, $x$, $y$ and $z$, so we say spacetime is four dimensional.
You might argue that time isn't really a dimension, but in relativity (both special and general) time is treated like space, and indeed time and space can get mixed together. So time really is a dimension just like length, breadth and width, and you can't separate time out on it's own.
You talk about infinite dimensions, but it's hard to see how spacetime could have infinite dimensions. In String Theory spacetime has ten dimensions, so a point in String Theory spacetime would have to be specified as $(t, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9)$ i.e. it needs ten numbers. But I don't see how you could have a spacetime that had infinite dimensions.
One possible confusion is that in Mathematics we talk about mathematical objects called spaces e.g. in quantum mechanics we have a Hilbert space and in many areas we have vector spaces. These can be infinite dimensional, but these aren't meant to be physically real: the word "space" is just slightly unfortunate terminology.
General Relativity doesn't actually specify how many dimensions spacetime has. The maths used in it would work just as well with ten dimensions as with four (though I don't think the maths would make any sense with infinite spacetime dimensions). It would also work with more than one dimension of time or even no time dimension, though this would result in a universe that looks nothing like ours. The number of dimensions is an experimental result i.e. GR with one time and three space dimensions predicts a universe that looks just like the one we observe.
Re your comment: I watched the video, and it's a nice explanation of spatial dimensions. However the presenter is confused about the time dimension. No-one since H. G. Wells has described time as the "4th dimension". In relativity we generally label the dimensions by a number, 0, 1, 2, etc, and we normally label time with the dimension zero. So when above I referred to a point in spacetime as $(t, x, y, z)$, we'd actually write this as $(x_0, x_1, x_2, x_3)$, where $x_0$ is the time co-ordinate. So I suppose we should say time is the zeroth dimension, though this is purely a matter of nomenclature and there is no obvious way to order the dimensions.
If you do calculations in relativity you do indeed have to consider time a dimension just like the spatial dimensions. The difference is a property that we call the signature. See http://en.wikipedia.org/wiki/Sign_convention for some info about this, though the article does go on a bit. Time has the opposite signature from the spatial dimensions. Some physicists give time a negative signature and spatial dimensions a positive signature, while others swap this so time is positive and space is negative. It actually makes no difference when you plug the numbers into General Relativity.
To give you a concrete example of this, in special relativity we define an invariant quantity called the line element, $ds$, by:
$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$
so we are adding together movements in time to movements in space i.e. we are treating them as the same thing. Note however that the movement in time, $dt$, gets a minus sign while the movements in space get a plus sign. The requirement that the line element, $ds$, be invariant is what causes all the weird effects like time dilation at high speed.
All of these answers are nice, but are missing the big perspective. Yes, there are 4 dimensions, 5 actually that we experience every day as long as a dimension is defined as a direction of movement possible while remaining stationary relative to others.
The 3 "normal" dimensions are easy to understand. An object can move forward/backward, up/down, left/right, and any combination of these directions (diagonal is the combination of 2).
Time is also a spacial dimension, and because of the mechanics of our biological systems, we only experience it moving in one direction at a constant rate. We could be moving "backwards" in time frequently, but because of how the mechanics of our perception work, we are only aware of time "passing" in the forward direction. From relativity, we also know that while time appears to move at a constant rate for everyone and everything, this is not true.
This leaves us with 4 dimensions that we experience every day. The 5th is also just as obvious, but only once you are aware of it. This dimension is called scale. For an example, take your computer screen. You now see it as a whole. Now without moving your eyes, focus on the comment box. No movement has taken place in any of the other 3 dimensions (we are always moving in time so it does not need to be counted in this case), and yet the scale of observation has changed. Now again without moving your eyes, imagine the single pixel exactly in the center of your vision. Again no movement has taken place in the other spacial dimensions, and yet our focus has "moved". How does this pixel work? There are indefinitely many interactions between microscopic particles going on in this smaller space. Electrons flowing, atoms moving about, all relative to other things at their own scale. Now imagine one atom in that pixel, still no actual movement has taken place. Is this atom by its self? No, there are millions of things making up this atom. Turns out, there is always a "halfway", and always a smaller and larger scale, and also that complexity doesn't change in relation to scale. It is therefor a spacial dimension, but one that seems abstract to us because of how we experience it and how it works. Things only interact with things at their own scale. Their pieces might interact with other pieces on a different scale, or the combination of things might interact with other combinations at a different scale, but the Earth interacts with the sun rather that a human interacting with the sun. On Earth we do so indirectly, but only because we are "part" of the Earth, which is interacting at its scale. Our body can't change its "scale of interaction", but our focus can.
One might say that this is false because of quantum mechanics. The basic premise in this is that the universe is divided into "quanta" or indivisible "spaces" that the smallest particles may occupy. The counter argument is simple. The idea of quanta allows computation of events at this smallest scale, where as without this understanding we would need to understand everything happening at infinitely smaller scales in order to accurately predict and calculate the behavior and trajectory of these tiny objects. It is possible to mathematically comprehend something with a finite number of variables but not something with infinitely many, so at some point one must say "things exist or they don't" in order to make an objective measure. The reason why quantum mechanics works and is still teaching us so much is that we picked something so small that we have no idea (and maybe never will) what something smaller could be, much less how to solve an equation with infinite variables. At this point, it is also useless to us to define smaller than a quanta (which stems from the word quantity).
Because of the definition of a dimension, there can in fact be infinitely many of them and we would never be aware of all of them. As long as things in the observable universe are constant with respect to all dimensions but the ones we are accounting for in any given calculation all equations still work, as no (perceivable) net movement has taken place in any others. In fact, it is entirely possible that our entire universe is actually just a 5 dimensional bubble in higher dimensional space, interacting with other things at its scale (un-observable to us) and making up part of an even bigger Universe.