If all diplomacy has failed and peaceful union is no longer possible, you need to move your units in order to stop the conflict with military means.
Most attacks occur automatically: If your army crosses path with a hostile army, or when its way passes the main city in a hostile province, fights will begin. If your army wins the fight, the remaining troops will execute the previously given command and commence moving into the original direction. Artillery also has an auto-bombardment feature: Idle long-range weapons will start firing at the closest enemy. It is often better to have units select their own target instead of giving an explicit attack order.
Even infantry units have a range. Thus it can happen that your army starts fighting before it reaches its target. In this case the fight will be finished before the army commences its way to the target.
In some cases more than one country starts an attack on a single enemy province. Should an ally of yours win the province before you do, your units will continue attacking the city and will cause a war between you and your ally. In order to avoid this, you should not attack the city itself in your initial attack order, but the troops stationed in the city. You can do that by selecting the army in the city as a target or, if the army units aren't visible, the flag that marks the city.
Are your troops on their way through a particular nation, they will cause war when they meet troops of the nation they are walking through. This happens, when their status to your country is either "war", "trade embargo", "ceasefire" or "peace". Peace just means that the particular troops won’t start an attack, when they’re in range (in case of weapons with range like Artillery, Railguns or Battleships). The only way for them, to cross roads in one of these statuses without triggering a war is while they’re on international territory (e.g. sea).
Nations which have a right of way, shared map or a shared intelligence can meet everywhere on the map without starting a fight.