Question: How do business rules relate to business processes?

First, be clear that rules and processes are not the same. The point seems obvious, but it's surprising how much difficulty many IT professionals have perceiving the difference. Indeed, if you've come up coding procedural programs or specifying use cases, seeing that rules are something different than procedural statements can be challenging, at least at first. So the Manifesto states the principle explicitly ...

2.2 Rules are not process and not procedure. They should not be contained in either of these.

The result of separating rules and processes is rule independence, a pervasive idea across the Manifesto's ten Articles. Its implications are far-reaching. For one thing, rule independence permits re-use of individual rules across all the processes and procedures of a business capability.

Although IT professionals readily 'get' the importance of 're-use', that is probably not exactly the right term to use for rules. If you were playing a game of chess or football, you wouldn't say, "we re-use individual rules any time we can". People don't naturally talk like that. Instead, you'd probably say something like, "we apply individual rules wherever relevant." In talking with business people and subject matter experts, we should be careful about wrapping what we say around implicit IT thinking.

Rule independence also provides a new, high-power lever for rule quality, something difficult to achieve when rules are embedded in processes or procedures. Just as for the rulebook of a game, rules for the business need to be cohesive - that is, not conflicting, misleading or incomplete. You also need to apply the rules consistently, so your processes get consistent results in like circumstances. The Manifesto summarizes these important points as follows ...

2.3 Rules apply across processes and procedures. There should be one cohesive body of rules, enforced consistently across all relevant areas of business activity.

References:

[1] The Manifesto is free, only 2 pages long, translated into 15 languages. Have a quick look (or re-look!). No sign up required. Well worth your time.