Why do we need units?
Because physics is an experimental science, measurements are required. You need some way to communicate results of experiments and measurements allow you to do that. These measurements are usually communicated quantitatively. Because you need to be able to reconstruct the results of experiments and make them legible to other people, some sort of reference measurement that you write other measurements in terms of is required. These reference measurements are called units.
What are fundamental units?
Some units are so fundamental that they can only be defined by how you can measure them. These are operational definitions. For example, a second is the amount of time required for 9,192,631,770 energy state transition cycles inside a cesium atom to occur when bombarded with microwaves. We define the unit by how we can replicate it.
Which system of units is used most often (in science)?
The main system of units used in the world today is the SI (International System aka metric). Each of the units in this system can be prefixed to denote some multiple of 10 increase. For example, a kilometer is just 10^3 meters
How do we use and convert units?
Every algebraic symbol in an equation represents both a number and a unit of measurement. When doing math with units, you can only add, multiply, etc. terms that are of the same unit. You must convert units if necessary. Also, once you're at a literal (aka a number and a unit), the units get treated like algebraic symbols and thus can be cancelled out and multiplied and whatnot.
We can convert one unit to another by getting the ratio of unit x to unit y and then (if necessary) multiplying that ratio by the amount of unit y we're trying to convert from.