What is 5 speed synehromesh gear box?

What is 5 speed synehromesh gear box?
A five speed gear box is an automobile transmission which has five forward ratios (speeds). Usually, "five speed" is used to indicate that the car has a manual transmission with a clutch pedal, however many automatic transmissions have five speeds as well.
A synchromesh gear box is a manual transmission which has synchronizers on most of, if not all of the gears. A synchronizer is a device which helps speed up or slow down the layshaft so that it can engage a gear smoothly when you move the gear lever.
In a car with a manual transmission, there are 3 pieces of the drivetrain moving at 3 possibly different speeds at a time: The Engine, the Layshaft, and the Driveshaft. The driveshaft connects to the wheels and is always spinning at a speed proportional to them. The engine speed is what you see on the tachometer. 
The layshaft is a shaft which transmits power rom the engine to the gears of the transmission. It is connected to the driveshaft through the gears whenever the gear lever is not in neutral, and it is connected to the engine whenever you are not stepping on the clutch pedal. This is why you can disconnect the engine and wheel speeds either by putting the car in neutral or by stepping on the clutch.
When you accelerate, the engine speed and layshaft speed are the same. When you shift into the next gear (commonly around 3000 rpm), you step on the clutch (disconnecting the engine) and slide the gear lever into the next higher gear. When that gear engages the layshaft, it needs the layshaft to be turning about 2000 rpm. The synchronizer helps slow down the layshaft to 2000 rpm and engage the gear smoothly. Without a synchronizer, you would either have to wait for the layshaft to slow down before engaging the next gear, or you would have to double-clutch: a technique where you use the engine to slow down the layshaft. 
All modern manual gearboxes are synchromesh gearboxes.
If a synchromesh gearbox is being properly shifted (with the clutch pedal fully depressed), and it is grinding as you engage a gear, it is likely that the synchronizer on that gear is worn out. if you are clever, you can still drive the car by double-clutching: take your food off the clutch in neutral, rev the engine to the speed it needs to be going for the next gear (yes you would have to memorize this), then step on the clutch pedal again and slide the gearstick into the slot. This method is also useful in non-sequential downshifts (i.e. fifth gear to second gear) as a way to abate wear on the synchronizer.