How did people apply for university in the 18th century?

Universities in Europe changed little during the early modern period and, in many respects, resembled the institutions of the late medieval period.

At the beginning of the sixteenth century, registration at a university was subject to rules derived from medieval tradition which were broadly the same in the whole of Europe. They.... varied little until the end of the 18th century...

Source: Hilde de Ridder-Symoens & Walter Ruegg (eds), A History of the University in Europe, vol 2

Concerning admission, this can be summarized as follows (from Ridder-Symoens & Ruegg):

• students could be admitted at any time of the year (even Sundays and holidays)
• basic formalities were “payment of a sum of money, swearing of an oath to abide by the academic authorities decisions and registration in a student roll.”
• the oath was essential, otherwise registration was null and void, but those under 14 were exempted from the oath
• “The amount due varied according to the social position and age of the students; nobles usually had to pay a larger contribution than the others, and those under 25 were entitled to a reduction, while the poorest, and sometimes those who had no intention of graduating were wholly exempt.”
• educational attainment required was usually vague but students needed enough Latin to understand lectures and to be able express themselves.

Students’ intellectual backgrounds very varied as some had had tutors while others were totally unprepared for the intellectual demands that universities would place on them. Consequently,

The universities of the period....used to cater for all levels of instruction, from elementary to higher, and welcomed pupils of all ages and degrees of knowledge......Until the end of the 18th century, then, no academic qualification was required for admission to the university, and there was no formal link between secondary school and university. From the 16th century, however, an organic and graded system began to emerge, assuming a more systematic and definite form in the Napoleonic period.

source: Ridder-Symoens & Ruegg

Varied educational backgrounds (and needs) also provided an opportunity for professors to supplement their income through private tuition. Salaries were often small and, in many cases, professors / teachers had to rely on class fees. Thus, when enrollment took a downward turn, many found it hard to make ends meet and some would transfer to other areas of teaching where student demand was higher.

Although there was usually no formal link, there were exceptions and grammar schools in England often served to prepare pupils for universities.

Concerning curriculum,

Until the end of the 18th century, most Western universities offered a core curriculum based on the seven liberal arts: grammar, logic, rhetoric,geometry, arithmetic, astronomy, and music. Students then proceeded to study under one of the professional faculties of medicine, law, and theology. Final examinations were grueling, and many students failed.

To these can be added natural philosophy, at first one of several branches of philosophy; the 18th century saw it "assume increasing independence from its philosophical origins". From the mid 19th century, it evolved into physics. (Acknowledgement: Pieter Geerkens for mentioning this in his comment)

Another key feature of European universities was their close connection to religion:

By the 17th century, both Protestant and Catholic universities had become overly devoted to defending correct religious doctrines and hence remained resistant to the new interest in science that had begun to sweep through Europe. The new learning was discouraged, and thus many universities underwent a period of relative decline.

With a few exceptions, this continued until the end of the 18th century.

Although some countries had begun to be secularized during the Enlightenment, most universities remained essentially ecclesiastical institutions, to the extent that they were either directly supervised by the respective churches or strongly connected to them through the importance of religious profession for the appointment of teachers, the admission of students, and the ideological orientation of academic studies and careers.

source: Walter Ruegg (ed), ‘A History of the University in Europe, vol 3’

R.D. Anderson notes that the 18th century showed

...the universities’ stagnation (particularly in France and in Great Britain) and their isolation from modern life, with much university teaching having become dubiously relevant to current social needs and intellectually obsolete.

As summarized by J.-Guy Lalande, in a review of 'European Universities from the Enlightenment to 1914'

One exceptions was Halle:

The first modern university in Europe was that of Halle, founded by Lutherans in 1694. This school was one of the first to renounce religious orthodoxy of any kind in favour of rational and objective intellectual inquiry, and it was the first where teachers lectured in German (i.e., a vernacular language) rather than in Latin. Halle’s innovations were adopted by the University of Göttingen (founded 1737) a generation later and subsequently by most German and many American universities.

All of the above refers to men (or boys in some cases). For women, there were far fewer opportunities and the first degree awarded in a professional faculty in Germany to a woman was at Halle in 1754. Nonetheless, the situation was improving:

...academic institutions in the 18th century gave the appearance of being less opposed than previously to the presence of women...a network of women’s schools founded during the 17th century for predominantly religious and confessional purposes was spreading across the whole continent.

Source: Ridder-Symoens & Ruegg