Here is a link to a glossary of botanical terms... you can look up simple definitions.
https://en.wikipedia.org/wiki/...
I have only studied roses in depth and am not a scientist. To give you a very basic overall look at how these terms are used, I checked my favorite rose book by Jack Harkness ... "Roses" ... and have kind of simplified the information.
Plants are divided into families with similar characteristics. For example the family 'Rosaceae' includes almonds, apples, apricots, blackberries, cherries, cotoneaster, geums, hawthorn, medlars, mountain ash, peaches, pears, plums, potentillas, quinces, raspberries, roses, sloes and strawberries and more.
So the "family" is broken down further into separate them further. Each of these subdivisions are called a "genus". The name of the genus for roses is "Rosa"
The next subdivision is into "species" ... which are members of the genus which grow true from seed.
In roses, and I am certain in other plant genera, species roses can cross-pollenate and create a new plant that does not grow true from seed. These are cultivars.
For roses, cultivars, often called varieties, can be created in nature or by man.
In the early days of rose breeding, breeders relied on nature to do the crosses for them and planted roses that they would like to make new roses close together in the hopes that a pollenator would take the pollen from one of the roses to another rose and create a new rose. They would plant the seeds to see "what came up". If a seedling looked good, they had to propagate it by tissue culture because it would not "grow true to seed". It was later in rose development that breeders made deliberate crosses and even longer before they kept records.
I hope this helps a bit.
Smiles,
Lyn