Ware shortage missions
You're asked to deliver a specific quantity of <ware> to a station, but this order won't appear in the Trade Window until the mission is accepted. The station in question will probably have a trade offer up for the same ware, but the mission will not complete unless you fulfil the mission's demand (this prevents you auto-completing the mission with a suitable trade that's already on it's way: you'll have to set up or cancel & re-set-up a new trade to fulfil the mission's requirements).
If you have (or can get) access to that ware for the mission's required amount through your own stations or discounts from NPCs (or simply stockpiling), take the mission first, then immediately set your tradeships to gather & deliver <ware> to take advantage of the trade bonus. If you don't do this immediately, the trade offers will update (as the mission committs you to fulfilling <ware> requirements, at a bonus) & you'll lose that opportunity..
(You can currently use <pause> as a slightly cheaty method to gain the maximum amount of profitsss from the latter scenario: <pause> allows you to scroll through & apply actions to a trade or property screen but doesn't update game events (such as movement, combat.. or trade offers!). If you close the screen you'll need to reactivate <pause> to gain access to another (or the same) screen)
Sometimes when there's an existing order for the same ware, the mission order won't appear in the trade window and you'll only see the existing order (normally for a different quantity). If that happens, try making a sale to the existing order and the mission one may appear. If that works, quickly cancel the trade via you ship's info screen; hopefully the mission order will now be present alongside the original (cancelled) trade order. HINT: select <mission> station's details from the map & look at their ware requirements: it's often easier than scrolling through the whole trade screen, even using the filter..
If the existing and <mission> trade orders are for the same quantity of ware, it becomes trickier. In such cases, it's probably best to try to fill both orders to be sure. This is another good reason for having a network of Trade Agents so you can check for existing offers for the ware before accepting a mission (you may gain rep but lose cash by accepting & completing a bad deal). At least the trade offer missions don't currently expire (they're not time-limited), but you'll likely lose profits by not fulfilling their /complete/ requirements ASAP.
Station Defence missions
S+M ships tend to arrive via highways or super highways, whilst L and XL targets boost in. All hostiles are hell-bent on attacking the station so best arrive ASAP or you risk missing the action.
In current implementation you don't need to participate in fight to receive success result. Stations capable of defending them self and you will receive reward even if you are in other zone.
Patrol Missions
In current implementation you don't need to participate in fight to receive success result. If you wait long enought, enemies will be killed by local patrols and stations and you will only need to visit mission target zones.
Assassination missions
Typically harder than station defence missions since targets tend to be located in positions in zones that are relatively far from stations. Makes boarding a more realistic option though.