What is spacetime (simple explanation)?

The intuitive and traditional idea of space and time is that objects live in an infinite three-dimensional box, space, and that their motion in space happen in time in such a way that at each definite moment in time all objects have a position, and we can compare those positions because time flows the same for all objects.

Physicists discovered that there is no such box, and there is no such flow of time. This traditional space/time framework somewhat holds but only relatively to an object; it is not the same for all objects.

So there is no universal spatial background, and no universal time flow.

Spacetime is then the notion we use to still have a background after all. By forming a space (in the mathematical sense) combining traditional space and traditional time in an intricate way allowing space to rotate into time and the other way round, we can still get by with the idea that there is some smooth universal scene where everything happens.

The price to pay to see spacetime as a background is that this scene is completely static, sometimes called the block universe. But since all space and all time are intrinsically part of it, it actually cannot be conceived from an external point of view, and indeed Einstein's equations are strictly local and relational: they describe how the distribution of energy defines its own playground and how time and space can be seen in the way we are used to only instant by instant for specific observers, whose mutual perspectives are always shifting and transforming. In that view spacetime is far from static, it is more like a sort of fluid.

An experimentalist's answer:

In particle physics the data show that one is not dealing with a three dimensional space that has as a parameter time, but that space and time are united in a specific mathematical way in what is called a "four vector space".

In classical mechanics a vector in space is a three dimensional row or column with values from the field of real numbers that follow eucledian tranformation properties under translations and rotation.

Momentum for example $(p_x,p_y,p_z)$ are the three vector components whose length is $p$ and is given by

$$p=\sqrt{p_x^2+p_y^2+p_z^2}$$

Which is invariant under rotations and translation of the reference system.

Relativistic mechanics which hold for high energies and momenta, and which are necessary in order to make sense of the plethora of particle data, requires four vectors,

The momentum vector above is modified and called a four momentum ( $c=1$ in this):

$$(p_x,p_y,p_z,E)$$ .

In this the "length" of the four momentum becomes the $\sqrt{E^2-p_x^2-p_y^2-p_z^2}$ and is the mass of the particle under consideration, an invariant in all Lorenz transformation frames.

So in analogy to calling three dimensional space , space obeying euclidean rules, we call the four dimensional space of this special construction, which clarifies the transformation rules in particle interactions, space time to denote that one is dealing with the special fourvector quantities.

Spacetime is, like the name suggest, space and time together.

But there's more to it!

3D space isn't just horizontal 2D plane and height together. You can also rotate stuff in it, so for some 3D object, you don't have uniquely specified what is its height - this can change as you rotate it. Similar thing happens with the spacetime.

In 3D space, when you rotate stuff, you don't do much to it, you only change coordinate system. Distances stay the same. Square of the distance $s$ is given by \begin{equation} s^2=\Delta x^2+\Delta y^2+\Delta z^2 \end{equation}

In the spacetime, you can obviously rotate stuff, but you can also do the "rotations" which involve time. They are called "Lorentz transformations" (they are not quite rotations, see gif). So, once you have spacetime, space and time can get mixed up. What remains the same is the following quantity: \begin{equation} s^2=-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2 \end{equation} where $c$ is the speed of light.

One might ask "So what? This looks like just a fancy math/set of definitions/whatever...", but there are some interesting consequences. Relativity of simultaneity is one of them. In simple words, it means that (contrary to some philosophies of time) there is no uniquely defined "now" (except "here and now"), just like there is no uniquely defined "this much to the right" in space. Consequence of relativity of simultaneity is that faster than light (FTL) travel could in principle be used for time travel. However, so far we don't know of any laws of physics that would allow for FTL transfer of information, but that's another story.