The Technic Pack wiki has an excellent article on the ins and outs of nuclear reactor design and terminology. Be warned that the article is outdated since the cooling methods in IC2 were changed (external passive cooling doesn't work anymore, and there are other coolant changes), but it's still an excellent article on understanding the basic design concepts.
Essentially, you want a reactor that is inefficient, so that it does not generate any more heat than it can dissipate passively by itself. This will run indefinitely without overheating – overheating is how a reactor goes critical and explodes. Such reactors are called "Mk I" reactors (say: "mark one"). How to make a Mk I is way too specific for a Q&A site, since there are as many designs as there are people trying to create more efficient designs. However, there is an online Java-based tool for simulating reactors, so you can test out designs safely: the IndustrialCraft Reactor Planner.
Reactors are classified by how dangerous they are:
- Mk I is a safe reactor that simply won't explode.
- Mk II generates a bit of extra heat, so it has to be turned off and cooled down occasionally. To get a Mk II rating a reactor has to be able to consume an entire batch of uranium (or more) before needing cooling.
- Mk III needs to be turned off to cool before it completes a full fuel cycle. To get a Mk III rating a reactor needs to be able to consume 10% or more of its fuel before needing to shut down and not damage any of its components.
- Mk IV is just like a Mk III except its design treats components as expendable, allowing some to melt before shutting it down. Before starting it back up the damaged/destroyed components need to be replaced.
- Mk V is for everything else. They are the most efficient at converting the uranium fuel into EU, but can only run for minutes or seconds at a time before needing to be shut off to avoid a meltdown and explosion. Carefully designed redstone circuits, or Nuclear Control mod stuff, are often necessary for these designs.
The more efficiently the reactor uses fuel, the more dangerous it is.
Breeders are an entirely different beast. They are designed to take depleted fuel and enrich it for use in a regular reactor. They produce relatively little EU and generally need to be run at a very high temperature. They can be safe, but the high temperatures means that miscalculations leave less margin for disaster control and require careful heat management built into the design.
Breeders are classified by whether they lose, gain, or maintain their heat level. A "negative" breeder needs heat added more or less often, usually in the form of lava buckets. A "positive" breeder gets hotter as it runs, needing occasional cooling or shutdowns. A "neutral" breeder has exactly-balanced heat generation and cooling, so it stays at a steady temperature. Sometimes a neutral breeder needs external heat (lava buckets) added to get to working temperatures, but after that can be left alone.
There are other classification details, but those are just about measuring efficiency.