Countries raise new land armies via recruitment. Recruitment costs money, and requires manpower. The primary source of recruitment is a country's own manpower pool. However, in Europe, mercenaries may also be hired.
How to Recruit
Once you've exposed the screen, you can see up/down arrows for each type of land unit. When you click the down arrow, one army unit's worth of the specific unit type is added to the number on the "Units Ordered" row; this is the proposed recruitment. Assuming you can afford the proposed recruitment, the "OK" button will be lit, and you can click it to initiate the recruitment operation. When you click OK, the recruitment screen will be closed, money spent, and an icon will appear on the map showing a blacksmith hammering away.
To hire mercenaries, click the "Recruit Mercenary Company" button. If you hover the mouse over it, you get a tooltip describing the composition of the company (how many infantry, cavalry, and artillery), and its cost. Mercs are a take-it-or-leave it choice: you cannot control the size or composition of the company. You can, however, wait a month; then sometimes a different company will be "first in line" to be hired.
The base costs for land units are modified by many of the domestic policies that a country has: aristocracy, offensive doctrine, land, quality, and serfdom. The resulting number is truncated to the next lowest integer. Finally, as with all costs, multiply by inflation. Again, the resulting number is truncated to the next lowest integer.
The cost of hiring mercenaries is not affected by domestic policy; inflation affects it. Mercenaries tend to be very expensive: they cost about double what their group would cost to recruit normally if the country had all middle slider settings.
For normal recruitment, there are several other costs for recruitment. First, it requires time to recruit. Infantry require 2 months, cavalry require 3 months, and artillery require 4. (If a unit is mixed, then it requires the longest amount of time.) Also, all normal recruitment draws men from a country's manpower pool. Once the manpower pool hits zero, no more troops may be recruited. (The manpower pool regenerates with time.) Finally, whenever a country recruits troops, whether at peace or war, its war exhaustion is increased.
For mercenaries, there are no other relevant costs to recruit. They don't require time to recruit or manpower, and they don't cause war exhaustion. As soon as you pay, a unit immediately appears (with low morale), and may begin operations. However, mercenaries are finite. There is a pool of mercenaries at any given time, and when the last group has been hired, no more mercenaries may be hired by any country, until more are generated.
The recruitment capacity of a city is the number of land units you can build there at one time. It is quantified in army units. Recruitment capacity is based on the province tax that a city generates: a city that pays N ducats per year in province tax has a base recruitment capacity of N units. That base recruitment capacity is potentially modified by two factors:
Any fraction is rounded down. If the result is zero, no armies may be built in that city.
Note that all of the normal modifiers to province tax (high stability, wrong culture and religion, revolt risk, etc.) thus also affect recruitment capacity.
You may occasionally notice that recruitment capacity in a city is one less than it ought to be. Here's why. The computation uses floating point binary numbers along with the truncation operator ("rounding down"). In such circumstances, the tiny computational errors inherent in the use of floating point can be amplified by the truncation operator. For example, a core province with base tax of 10 and net province tax multiplier of 70% ought to be able to recruit 7 army units at a time. But what really happens is that 70% cannot be represented by a non-repeating binary number and so it is represented internally as number very near to 0.7, but a tiny bit more or less. For the sake of this example, assume that 0.7 is represented in floating point as 0.69999995. So the province tax is 6.9999995d. You will never notice the missing 0.0000005d in taxes. But in the computation of recruitment capacity the number is truncated, allowing recruitment of only 6 army units instead of 7.