There’s a lot of weird systems out there, and none of them were always strictly adhered to, as there are many cases of monarchs trying (with mixed success) to establish a recognized heir in contravention to the common practices of their realm.
Salic primogeniture was very common in Europe and among many monarchical societies, especially prior to the Renaissance. Under salic primogeniture females simply cannot succeed, they do not have any rights of succession whatsoever, and, succession can’t be traced through females either in many cases.
Semi-salic primogeniture was also common, and in practice females could not succeed under that system either.
An example of the difference between salic and semi-salic primogeniture:
King William has three sons (Henry, John, Richard) and one daughter (Mary.) He also has a younger brother (Edward.)
If all three of William’s sons die before him, and if none of those sons had sons (or if their sons had died, too) then succession is traced through Edward’s line. If Edward himself has died before William, then Edward’s son succeeds.
Under semi-salic primogeniture, in the same situation, before succession is traced through Edward it is traced through any sons of Mary’s, meaning Mary’s sons could inherit, and they would have precedence to inherit over Edward, but Mary herself could not succeed.
In many cases it’s obvious why salic primogeniture is preferable here to semisalic primogeniture. Daughters were typically married off to other nobles, sometimes other royal houses. If William, who is the King of Scotland had married his daughter Mary off to someone in the French royal family who happened to become King, then upon William’s death the King of France’s son stands to inherit the Scottish crown (and the French one–eventually, obviously.)
A lot of different terms are used interchangeably and with different meanings when it comes to labeling laws of succession. What I call salic or semi-salic has different names according to different people. What I refer to as salic primogeniture is roughly synonymous with what is called agnatic primogeniture and what I call semi-salic primogeniture is roughly synonymous with what is called Quasi-Salic succession.
The British system is referred to both as male-preference primogeniture and agnatic-cognatic primogeniture. Basically it means succession goes to the oldest male first, but the oldest daughter can inherit if there is no oldest son, for example.
Full cognatic primogeniture is a system by which the oldest child, regardless of gender, succeeds; and is seen in Sweden I know for sure and possibly some other European monarchies recently.
Primogeniture definitely became the norm in Europe but it was not always so.
In pre-conquest England there was a land inheritance system called “Gavelkind” in which land was divided up equally among all the male heirs. This system also allowed land to be willed to a successor as well. Similar systems were seen in Wales, Ireland, and Scandinavia especially during the Dark Ages. Salic gavelkind would equally divide the land between the male children, semi-salic gavelkind allows inheritance to also be traced through the female children, meaning a son of a daughter would also be entitled to some inheritance. Typically if your inheritance traced through the female line you only received half a normal share.
Gavelkind may even be more properly classified a a type of salic patrimony, which was itself just basically a system which divided land up between male heirs. This was the norm especially in clannish societies in the more “unsettled” parts of Europe during the dark ages (or middle ages if you prefer.) The Frankish kingdom was divided between male heirs in this way.
The benefits of a gavelkind type system where you divide up lands between the surviving sons is obvious, there is less chance of brothers warring against one another for the crown if all the brothers get some piece of the pie after the father’s death. The downside is this tended to fracture a realm more and more to the point where you could not meaningfully look at it as one united kingdom. France/Germany were both very fragmented politically and regional nobles for many centuries wielded far more power than any centralized monarch, who was often little more than a figurehead or just simply the most powerful noble amongst a group of peers. Norway was also likewise fractured by non-primogeniture type inheritance laws from the 1000s-1100s A.D.
Elective succession tends to be highly specific to the realm/kingdom/country in question. The Holy Roman Emperor was elected by electors. Rulers of certain states within the Empire had voting rights, rulers of lesser states were typically not electors.
The HRE was more of a confederation and its ruler was more of a figurehead as a leader of the loose confederation. Although once the Habsburgs more or less “acquired” continuous succession of the Empire they clearly became the dominant noble house in the HRE and wielded a great deal of influence within it, but still did not truly rule over the entire HRE as though it was a unified state.
Poland-Lithuania’s elected monarchy was a bit more democratic, instead of just a few of the most powerful nobles having elector rights (as was the case in the HRE) every noble that bothered to assemble for the election could vote. During the Jagiellon dynasty there was sort of an elective system, but Poland-Lithuania was in practice ruled by a single dynasty during that time, when the last Jagiellon king died with no heirs to speak of Poland-Lithuania established a truly elective monarchy in which all the nobles in the realm had a right to decide on the future monarch. Usually several thousand nobles cast votes, and consequently the monarchs of Poland-Lithuania were somewhat weak. Eventually outside powers even began to exert influence in the elections to put up puppet rulers when Poland-Lithuania became one of the weaker states in Europe.