There was not a single set of tracks from one end of the line to the other. There are specialized side tracks placed along the route specifically to permit two trains to pass. See configurations H, I, and J in this image.

Whichever train that was scheduled to reach the siding first was to pull off the main line to allow the other train to pass. This is why the development of accurate time pieces and time zones was vital to the advancement of the rail system. Both were needed to accurately determine the location of trains on the line to avoid collisions.

hi! not discouraging discussion here, but just fyi, this isn't necessarily a historical question: there are single tracks all over the place. You can learn how they work from rail- or transport-related subs, e.g. /r/trains, /r/railway, /r/railroad

There are couple of way that this would work. As /u/brettmjohnson said you had places were one train could pass another, most commonly at stations or halts. This is very common in British rural railways, especially along the Welsh Coast. One line for the station, on for overtaking.

However, in the 1860s to early 1960s, at least it Britain, there were many branch lines that only carried on train at a time, on both the up and down line. At least to me, the most local one was the Wellington - Coalport line, The Coalport Branchline. It was a single track line for the majority of it's existence, and would only have a single loco travel along it, The Coalport dodgers as they were known. Even the Stations only had the one track as can be seen here at the Madeley Station. Being a rural and industrial route, it had limited travel, about 6 return journeys a day. Single track was not a problem. The train got the end and could switch around using points, and take the train back reversing.