[speedfreek]

The figure of 78 billion light years is known as the comoving distance.

The furthest, oldest objects we can see are around 13.5 billion light years away in each direction. So that is the radius of our visible universe. But... we see those objects where they were 13.5 billion years ago. Due to the expansion of the universe, they are now estimated to be something around 78 billion light years away (the comoving distance).

The narrator got it wrong though, he said the universe is estimated to be 78 billion light years across, then he should have said 78 billion light years in radius.

And this all just refers to the visible, or observable universe, which is all we can know about due to the speed of light and the age of the universe. It could actually be a lot bigger than that.