Measurement is fundamentally an act of comparison: to say a rope is '10 metres long' is to say it is 10 times as long as an object whose length is defined to be exactly 1 metre -- and '1 metre' is the unit of length.
An arbitrarily chosen standard of measurement of a quantity, accepted internationally, is called the unit of that quantity.
Units mirror the fundamental/derived split of physical quantities: fundamental (base) units measure the fundamental quantities directly (metre, kilogram, second, ...), while derived units are built by multiplying/dividing powers of the base units (m s−1 for velocity, kg m s−2 for force).
A system of units is a complete set of units covering every fundamental and derived quantity. Historically, mechanics used three competing systems, distinguished by their choice of base units for length/mass/time: f.p.s. (British Engineering: foot, pound, second -- the one non-metric system), c.g.s. (Gaussian: centimetre, gram, second), and m.k.s. (metre, kilogram, second).
Since 1960 the internationally agreed system has been SI (Système International), recommended by the General Conference on Weights and Measures (GCWM) in 1971. SI is rational (exactly one unit per physical quantity), coherent (every derived unit follows directly from the base units, no fudge factors), and fully metric (multiples/sub-multiples are always powers of 10, using standard prefixes from deca/101 up to yotta/1024, and deci/10−1 down to yocto/10−24). SI's seven base quantities and units are length (metre), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), amount of substance (mole), and luminous intensity (candela) -- each pinned down by a precise physical definition (e.g. the second via 9,192,631,770 periods of a Cesium-133 transition).
Plane angle (radian) and solid angle (steradian) were reclassified from 'supplementary' to derived quantities by the GCWM in 1995. One radian is subtended at a circle's centre by an arc equal to the radius (π rad =180∘, so 1 rad ≈57.27∘); one steradian is subtended at a sphere's centre by a surface patch equal in area to the radius squared.