Big old answer but hopefully a little entertaining
Classical physics has the idea that electrons are solid particles passing through the two slits. If this were the case, you would expect to see on a screen behind it that there are two distinct regions where around half the electrons went through the left slit and half through the right shown here:
However what actually appears in experiments is this:
What the hell is happening there? Good question. Where else in nature is that sort of pattern observed? Ah yes, water waves from two sources interfering with each other:
So you might be a happy physicist if you call it a day there and say 'OK our particle model of an electron doesn't seem to be entirely accurate and because I've just seen it behave like a wave like in that great video above, I would like to go home now'. And leaving the experiment at that, this experiment is one of a few examples of what is called wave-particle duality.
But just saying electrons behave like particles in one situation and waves in another isn't really explaining what might be happening. We can go further...
The next day when the physicist turns up to work again, he tries something new. Instead of sending a constant stream of electrons through, he sends one at a time. He thinks he's being clever because now the electrons cant interfere with other electrons passing through other slits, however as he records where each electron hits on the screen, he finds that the same interference pattern emerges like before. The physicist goes home because he is annoyed at the result. 'How the hell does an electron, a solid particle passing through one or the other slit create an interference pattern when you add up all the positions where they landed?!' the physicist said to his cat. The cat replied 'Meow have you tried measuring which slit the electron is passing through meow?'.
He comes back the next day and does as his cat suggested, because if he knows which slit the electron is going through, he can surely determine where the electron will land. The result? This:
The way that physicists have gone about modelling this, is that electrons in the case of the double slit experiment are behaving like a special kind of wave where the height of the wave represents a higher probability that the electron will be found there. The electron isn't splitting when it goes through the double slit and interfering with itself, its more like wave of probability is passing through the slits and the interaction withthe slits makes the probabilty wave look like this:
And then when it finally hits the screen, the wave is forced to collapse into a definite location/point and there will be spots where the electron is more likely to be recorded.
