You can buy units from a city info menu - but only if you control the country's capital.
Above the list of units you can see how many are available at the moment - this number is determined by the city's population. You get reinforcements in the beginning of the game, and then once a month (every 4 turns).
There are 3 types of units: land, naval and air. Naval units, obviously, are only available in ports. To see more information about the unit, hover the mouse over the unit image. To move all units of this kind, click on the image.
Stealth units, like Submarines, Marines and Stealths are invisible to your opponents until they they are detected or participate in a battle. After a battle you need to take them to any of your cities to restore their stealth.
All units cost a certain amount of money to purchase, and afterwards you'll have to pay 1/10 of this sum every week for their maintenance.
Hold the '+' button to buy all available units. If you haven't moved the purchased units yet within one turn, you can still sell them back by clicking the '-' button.
On the map, the cities with available units are highlighted with thicker outline.
There's a direct correlation between city population and reinforcements. If your city population falls down to 50%, only half of the reinforcements will be available.