What is the long run equilibrium in layman's terms? My textbook states
An economy's long-run equilibrium is the position it would eventually reach if no new economic shocks occurred during the adjustment to full employment. You can think of long- run equilibrium as the equilibrium that would be maintained after all wages and prices had had enough time to adjust to their market-clearing levels. An equivalent way of thinking of it is as the equilibrium that would occur if prices were perfectly flexible and always adjusted immediately to preserve full employment.
Does this mean that, in the long run, ordinary people like me have already found jobs with our wages already adjusted to price levels affected by inflation and monetary policy by the Fed? In comparison to the short run, is the transition period where the price of money fluctuates with increasing/decreasing money supply by the Fed non-existent? What is the long-run equilibrium for the average Joe and how does this affect me personally?
Long-run equilibrium is a theoretical construct. One may argue that an economy never really gets a chance to actually reach it. So it is best to explain it in a hypothetical setting:
Imagine an economy with fixed population and all markets in equilibrium. By all markets I mean the money market (no excess money, no shortage of money), goods market (no pilling of goods in factories, no shortage either - producers' plans for production are unchanging) and labor market (whoever wants to be employed at the going wage-rates is employed). Now, strictly speaking, one more market, bonds market, is left, but by the famous Walras law, when $n-1$ out of $n$ markets are in equilibrium, so will be the $n^{th}$ one.
You can say that the economy is currently in long-term equilibrium. It is producing at full capacity. So let's give it a demand-side shock. It could be a monetary shock by the central bank, or by government by increasing/decreasing spending, external shock due change in export demand, etc. Basically, a shock which shifts the aggregate demand curve.
Now we have a disequilibrium. Not all goods that are produced could be sold (or not enough are produced). There is more money circulating in the economy than required (or not enough money to meet the need of increasing transactions). What do we do now?
Usual option: In the immediate aftermath, we should produce less (or more) to meet the new demand. To do this workers are fired (or more are hired at higher wages) changing the labor market equilibrium as well.
Alternatively, we could also just slash (or increase) prices so that demand is balanced again. But there are problems with this. It is almost impossible to do this in a synchronized way everywhere. On the top of if, producers would also want to simultaneously cut wages (or workers will demand higher wages due to higher prices - that producers may not be immediately agree to). This alternative option, as per macroeconomic literature, is not smooth. It faces rigidities and happens slowly. As your book points out, if, hypothetically, it did happen then there would be no change in employment or output levels.
This is what distinguishes short-run and long-run equilibrium in macroeconomics. The usual option is the one that creates a short-run equilibrium where the prices and wages are yet to be adjusted to reach the long-run equilibrium.