Monthly movement is a method of reducing movement attrition, by moving armies (or, sometimes, fleets) only during a month, but never being in motion at month's end.
In EU2, attrition is computed only at the start of each month. But if you look at land movement times, most single-province moves can be completed within a month. (Cavalry can do many double province moves during a month.) If a move can be completed in 29 days or less, then it can be done during a month. If it takes 30 days or more on land, then it must necessarily overlap a month end and thus cannot be done without attrition. By sea, movement attrition can be avoided even for moves of multiple sea zones lasting more than 30 days.
Thus, the typical way to play solo is to freeze the game on the first of every month, then make all movement orders for that month. Now you run the game, dealing with anything that comes up during the month. Most moves will complete during the month, thereby causing no attrition.
An army which is led by a conquistador does not take any movement attrition. Therefore your conquistadors should be continually moving.
Another exception is tiny armies, which don't need to worry about movement attrition, because they take fractional attrition and therefore can move for 99 month-ends before being eliminated. Also, they are cheap, so you can usually replace them easily.
Naval attrition is somewhat different, in that ships are much larger quanta of force, and attrition on them can be cancelled if they enter a port. Thus, most medium sized or small fleets can move all the time without taking any casualties. However, if a fleet is carrying passengers, then the passengers take movement attrition, and again you want to move the fleet during a month.
Fleets have two exceptions, though, where you never need to worry. One is national waters. All fleets (and passengers) in your own national waters take no attrition of any kind, including movement attrition. The other is ports. When a fleet (again, including passengers) is in a port, including when it is in motion coming out of port, it takes no attrition. Note that if you stop in the port, then the passengers will unload, but you can order fleets in and out so that they don't stop.
For example, outside of northern sea zones, galleys take 20 days to move to a new sea zone and 10 days to make a round trip in and out of port. If a fleet which includes galleys starts movement on the first of a month going to an adjacent sea zone, and it goes in and out of a port where it has military access, it takes no attrition. The fleet will be in port on the last day of the month but will emerge from port on the first and so will be considered in port during the end of month to first of month date crossing.
Unlike land units, warships and transports speed up with technology. See the article on naval movement time. At increasing naval tech levels, the range a fleet can move within a month will increase, from 1 sea zone, to 2 (at naval 9), 3 at naval tech 17, etc. As naval tech improves, it becomes increasingly easy to plot long moves for your fleets which always end in national waters or in port.