Assignments are a source of income, they will earn you NIC, tokens and in some cases charges or equipment. Completing industry assignments will increase your relation with the faction you're working for, which will increase your efficiency with all facilities (factory, refinery, recycling plant, repair shop, etc.) in terminals from that faction.
Assignments can be obtained at any terminal or outpost. The assignment levels available depends on your relation with the faction's terminal you're docked at. Your can find your relations under Agent profile > Relations.
With the Parallel Assignments extension (under the Relations category), you can accept one more assignment with every level.
Repeating the same assignments too often will hide it (temporarily). Basically, 6 out of the 10 last assignments cannot be the same.
Completing assignments will increase your relation. For example, a level 1 mission will allow you to increase your relation to 2.01 without issues, and it will slow down up to 3.01. To increase beyond 3.01 in this example, you will have to move on to level 2 assignments (which are accessible when your relation is at 2.01 or higher).
When your relation increase seems to be 0 while you theoretically should be able to increase your relation, it's probably due to a rounding error.
Increasing your relation with one faction will not damage your relation with other factions.