The London Underground is a public transport network of underground railways in London, also called The Tube.
Dumbledore is referring to the Tube Map, which is the map used by travellers on the underground. It’s not simply an overlay of the railway lines onto a map of London; it’s a simplified version of lines to make it easier to follow. It’s become a cultural icon for London, and is recognised as a design classic. This is how Wikipedia describes it:
Beck [the designer] was a London Underground employee who realised that because the railway ran mostly underground, the physical locations of the stations were irrelevant to the traveller wanting to know how to get to one station from another — only the topology of the railway mattered. This approach is similar to that of electrical circuit diagrams. […]
To this end, Beck devised a simplified map, consisting of stations, straight line segments connecting them, and the River Thames; lines ran only vertically, horizontally, or on 45 degree diagonals.
This is a recent version of the Tube Map:
It’s a very distinctive image. Most people in Britain would instantly associate it with London. It’s quite an unnatural arrangement of lines (and hence quite remarkable), but also something that most British readers would be familiar with.
I expect it was intended as a joke for British readers. Rowling probably didn’t expect the original books to have such a large international audience.
(Sorry if it that seems as I’m over-explaining it, but I’m guessing that since the OP isn’t a native English speaker, they probably didn’t grow up in the UK and don’t realise the map’s cultural significance.)
As far as I know, it’s never explained how he got this scar. Rowling said in an interview that she might explain it eventually:
How did Dumbledore get his scar in the London Underground?
You may find out one day. I am very fond of that scar.
— J.K. Rowling at the Edinburgh Book Festival (August 2004)
But I don’t think she ever followed up. Perhaps a future Pottermore update?