The SI Base Units
Every number in the whole of science can be represented with the units derived from 7 base units. Any more than that is unnecessary: for example if you have defined both length and time then you already have everything you need to define a whole lot of other units. Area is length times length, velocity is length over time, etc. Through in another base unit such as mass and you can then define force (length x mass / (time x time)) momentum (length x mass / time) and so on.
Dimensional analysis of science shows that seven base units are both necessary and sufficient to describe our universe. In practice you can pick any seven units that aren't just linear combinations of each over. For instance you could get rid of time and mass if you put a unit for momentum and force instead. The units we have picked are metres (m), seconds (s), kilograms (kg rather than g. I know it's weird), candelas (cd), Amperes (A capitalised because it is a name), Kelvin (K, ditto) and moles (mol).
All of these had some original thing it was picked to be defined as base on a property of some arbitrary object. As time has gone on these definitions have changed and now only the kilogram is based on a physical object. Let's start with each of the original definitions to get an idea of where they started.
Metres were originally 1/10,000,000th of the distance between the equator and the Geographic North Pole as measure though Paris (it was the French that standardised the usage). Knowing that the distance is 10,000 km is quite useful when estimating distances between countries on the map.
From that we defined a kilogram as the mass of a litre of water. Notice that since a litre is a thousandth of a cubic metre we needed the previous definition to define this one. However seconds were defined simply by dividing the day into 86,400 equal parts which doesn't require any previous definitions.
Current required both of these in its definition. An Ampere was the amount of current needed to deposite 1.118 mg of silver per second from a silver nitrate solution.
Kelvin didn't require any others. The magnitude of a Kelvin is one hundredth of the temperature difference between the triple point of water and the boiling point of water (just like Celsuis). Then it sets it's 0 at absolute zero (so it requires three different physical measurements to be taken.
Moles are just a mass over another mass with a constant thrown in, so they are kind of unitless dimensionally speaking. Finally candelas measure luminous intensity and were compared to how many candles you would need to have to produce the same light output. They’re definition started off particulary ill-defined.
Over time we had problems with all of the measurements these things were based on. If you measure the length of the distance between a pole and an equator to a higher accuracy or you find that a previous figure was wrong then you are constantly working with slightly shifting units for everything. There were two solutions: either you could make a physical object to a high specification and have that as the immutable definition, or you can define all of these as ratios of physical and mathematical constants which will never change. On the whole we mostly tried method number 1 for each and then slowly switched to method to number 2 at a later period.
From the 1960s onwards only the kilogram is defined by an actual object, so we'll begin there on the modern definitions. In a vault in Paris lies a cylinder which is made of a platinum-iridium alloy. The mass of this cylinder is The Kilogram (capitals intended).
The second is the first to be defined as just a physical constant: the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of a caesium 133 atom. These corresponds to approximately the same as just dividing the day up as before, but gets around defining awkwardness such as leap seconds, leap years, sols vs sidereal days and the Earth slowing it's rotation. (Which are all articles in their own right on here.)
At this point we can refer to this chart which shows which units are defined from which other units:
So seconds and kilograms are defined from a physical measurement and an actual object, but most of the others require at least one of the others. That means that if you were to change the definition of a kg slightly then candelas, Amperes and moles would change too. The only other independent unit is Kelvin, which has the same definition as before with some small specification changes based on the exact composition of the water that the triple point is defined from.
The other four are all defined from others. The current definition of a metre is the distance traveled by light in 1/299,792,458th of a second. This comes from the speed of light which we know to be safely constant. A mole is defined as the number of objects which matches the number of atoms in 0.012 kg of Carbon 12 which are at rest and in their ground state. A candela is the luminous intensity of a source that emits monochromatic radiation of frequency 540 terrahertz and that has a radiant intensity in that direction of 1/683 watt per steradian. Think of a steradian as an angle but in a 2D.
Finally an Ampere is the current needed in two infinitely long wires which are placed 1 metre away in parallel to create a force of 0.0000002 Newtons per metre. This equates to the base units kilograms per seconds squared. Both this and the candela and the Ampere needed all three of metres, seconds and kilograms defined first.