Maintenance

You could just go the kerbal way and wait for a problem to happen, sure: but if you'd rather avoid surprises, you can perform preemptive maintenance and save you some trouble later.

All the following activities can only be performed during EVAs, by flying to the part and activating the corresponding button.

You will need a kerbal that is qualified to work on that part: refer to the page about perks.

Inspections

You can inspect a part before an important manouvre to ensure that everything is in order.

When you inspect a part, it gains a short and temporary reliability boost. Specifically, the chance of failure is set to zero and then grows back linearly to 100% over the course of some seconds.

This is particularly useful for engines: you can check them out, and then perform your burn (more) safely!

Maintenance

You don't need to wait for a part to break, you can perform pre-emptive maintenance early on to give it a permanent boost, that will decrease the part's age by a considerable percentage.

However, maintenance is not free: you will have to spend some spares. However, the cost of maintenance is usually a lot lower than the cost of repairs...

Repairs

If (when) a problem happens, it's not the end: you can still get out and repair the failed part.

To do this, just:

1. go on EVA;
2. fly to a container of spare parts (typically, your pod);
3. fly to the broken part;
4. activate the repair button;
5. profit!

Additionally, a smart kerbal might be able to figure out a smarter way to repair the damage, and save some spares.

After a repair, a percentage of the part's age is discounted, because the broken component has been replaced by a newer one.