Basics of attrition
In my first game I went to war a lot, and therefore I was very surprised when I found that after 150 years my losses caused by attrition where almost double the losses in combat. It turns out this is historically right. Bad sanitary conditions and low or no pay were the biggest army killers through deaths, medical leaves and desertion. Attrition is probably the biggest way of wasting resources in EU2, and therefore your worst enemy. Attrition is coded in a funny way in EU2 and requires a different approach whether an army or a fleet. You see the attrition in a small skull in the army/fleet information screen. That skull can be green, yellow, gray or red, and has a number that often times is slightly incorrect. Sometimes is 1% more than showed due to movement attrition not being correctly displayed. One vital piece of information is that attrition will be applied on the first day of the month, as if the troops went all in medical leave or deserted after getting their paycheck. Although attrition happens both at war and during peace, you will probably suffer it a lot more during war times and while exploring unknown territories.
Any army in movement suffers 1% attrition, so don't move armies around for the sake of it, and consider sea transport when the distances are big, and/or your armies have to move through other nation territory and/or during winter. An army inside a moving fleet also suffers from movement attrition, but they move much faster. An army is only counted on moving the first of the month, so if it starts moving after the first and stops moving before the first of the next month, for attrition purposes it hasn't moved. The only way to reduce movement attrition is through leaders that have a high movement rate. Movement attrition is not added to the supply limit attrition, you will only get the higher of both. Conquistadors eliminate movement attrition.
Any army will suffer supply attrition when its number of units exceeds the supply limit value of the province they are at the end of the month. The actual supply limit (ASL) is based on a base value that is displayed and several modifiers that are not, and is very difficult to know, but you can have an estimate. The displayed supply limit (DSL) is the number that appears in the province screen. It is always based on the base tax value (BTV) of the province. If you own the province (even if you don't control it) the DSL is 5x the BTV increased with each fortification level and also if a conscription center is present. If the province is owned by an ally it will be 2x BTV and will become 5x BTV if the ally fights in the same war. If the province is owned by another nation the base value will be 2x BTV, 3x BTV if you control the province or 4x BTV if you have military access (even if at war with them). In all cases the base supply value is doubled if you carry a naval blockade (only available for provinces with a port), but this bonus ceases to apply the moment you control it, or during both land or naval battle that take place during the siege. This means that you should try to break the enemy blockade at the end of the month, even with a hopeless naval battle, so the troops in the siege will lose the bonus on the first of the month and suffer more attrition, during the rest of the month the blockade is irrelevant in attrition terms, although not for siege advancement.
The other number in the province information screen is the maximum attrition that you can suffer in that province, it goes up in difficult terrains, harsh climate, when the province has been looted, and when the province is out of supply for you (the DSL number is red). It also has a tech level penalty that goes down with every sixth level that you reach. The higher the max attrition number, the more troops you will lose.
The ASL will be the DSL modified for your armies, increased by 2x the movement factor of your leader (4x for a conquistador) and reduced by 5% for tropical climate and 10% for winter, that is increased to an awful 25% if you don't own that winter province. Climate therefore increases attrition while reducing supply limit. So to make your estimates, look at the DSL, add 4 for the default leader, subtract the penalties and you get the maximum number of units that will not suffer attrition. So if attacking a 20 DSL province in Russia, you will go from an ASL of 24 units to zero when winter comes, and you will suffer the maximum attrition, which can be very high when at low tech on a looted province in winter and if you don't have a supply line. This in practical terms means that your army will become irrelevant or disappear in a very short time. Napoleon didn't know as much as you do now.
The DSL will be green in provinces that you or your war allies control, and it will be yellow in provinces that you can reach while keeping a line of supply to a province controlled by your war alliance. All other provinces have a red DSL meaning out of supply.
The army attrition information is given by the number near the skull. The color of the skull is green (yellow in tropical areas), and turns gray (but stays yellow in tropics) when there is attrition. It will be red when in a foreign province.
The number near the skull can be 1% less than the actual attrition.
Infantry takes 5/6 of the attrition and artillery doesn't take any attrition as long as there are other units (artillery does get siege attrition though). That's why at the end of the war your armies are made mostly of artillery with some cavalry. Plan on replenishing your infantry continuously so your more expensive units don't pay the attrition price.
One last important aspect of attrition is that very small armies that in theory should suffer very little attrition when out of supply, get a minimum attrition penalty, that goes from 1% when between 500-1000 soldiers and go over 10% when smaller than 100 soldiers. If you arrive first to a siege and your allies start to pile armies until they go over the supply limit, you should take your army out, leaving a small force to take control of the province when it is conquered. If your force is too small, attrition will increase quickly and your force may not survive to the end of the siege. Since reorganization is not allowed during a siege, I recommend that you break your army before moving into the siege, and move out the larger force when you see that over-piling is going to happen. Your other option is to cover the province, that leaves about 1000 men per fortification level.