Definition
Local mean time is the time of day at one place, based on when the sun reaches its highest point there, averaged over a whole year. Even a small change in longitude changes the value, so each city had its own clock time. Until standard time took hold, this was the correct time in each town.
Clocks in neighboring cities differed by a few minutes
In the 19th century, just before the railroads arrived, people set their watches by the big clock in the town square. That clock had been set by the sun in the sky. Someone would stick a pole in the ground, watch for the moment its shadow was shortest, and then add a correction that changes with the seasons to set the clock. We will look at why that correction exists in the next section. All you needed was a pole and the sun, so any town could do it. Long before that, people simply used whatever time a sundial showed. From the late 18th century into the 19th century, cities switched over to mean time one by one.
But the moment the sun is highest is different in every town. Towns to the east reach it first, and towns to the west reach it a little later. The difference is one hour for every 15 degrees of longitude, or 4 minutes for every 1 degree. So when you arrived in the next city and looked at its clock, it would be a few minutes off from yours.
The important thing is that nothing was broken. Each clock showed the correct time for its own city. If a city a full day's trip away ran a few minutes different, it caused no trouble at all.
Then trains and the telegraph came along, things that moved faster than people, and those few minutes started to matter. To write a timetable for a train that passes through several cities in half a day, you first have to decide which city's time the departure time refers to.
Where the word 'mean' comes from
The moment the sun reaches its highest point in the sky is called solar noon. But the time from one solar noon to the next is not the same every day. Earth's path around the sun is an oval, not a circle, and its axis is tilted, so that length shifts a little with the seasons.
If you used it as is, the length of a day would keep changing, and building a clock would be hard. So people averaged a full year of days and used that average day as the unit. That is exactly where the word 'mean' comes from.
There is also a kind of time that treats each day's actual solar noon as noon. It counts the moment the sun is truly highest as noon, and it is called apparent solar time. Local mean time takes those moments and averages them over a year, while apparent solar time uses each day's moment as it is, so the two are not the same value.
The two values run ahead of or behind each other at different times of year. This gap is part of why a sundial and a wristwatch sometimes seem to disagree.
A bit more precisely: how is it different from standard time?
Local mean time is based on the longitude of the spot where you stand. Standard time is an agreement for a wide region to share the time of one chosen meridian. One changes from place to place. The other has a single value across a large zone.
So you can't really say standard time is a worse or more 'off' version of local mean time. They simply use different references. Local mean time fits one spot exactly, but it disagrees with the next city. Standard time lets a wide region use the same number, but it drifts a little from the sun.
Local mean time hasn't disappeared, either. People still use this scale when working out when the sun is highest at a given place. The same goes for astronomy and navigation, where position matters.
What we don't do is set everyday clocks by it. The numbers we read are the ones a wide region agreed on together, and local mean time remains underneath as nature's own scale.
Common misconceptions
Local mean time counts the moment the sun is actually highest as noon.
That describes apparent solar time, which uses each day's solar noon as it is. Local mean time averages those moments over a year, so it drifts a little from each day's solar noon depending on the time of year.
Cities had different clock times because clocks back then were inaccurate.
Each city's clock showed the correct time for that city. It wasn't about how good the clocks were. Each city simply used a different spot as its reference.
Where you meet it
In the 19th century, just before standard time came into use, townspeople set their watches by the big clock in the square. What that clock showed was the town's local mean time.
Work out the solar noon for two cities at different longitudes and compare them. The gap between them is the difference in their local mean times.
In one sentence
Local mean time is a place's solar noon averaged over a year, so it changes with longitude, which is why each city had its own clock time.