What's the best practice for primary keys in tables?

I follow a few rules:

1. Primary keys should be as small as necessary. Prefer a numeric type because numeric types are stored in a much more compact format than character formats. This is because most primary keys will be foreign keys in another table as well as used in multiple indexes. The smaller your key, the smaller the index, the less pages in the cache you will use.
2. Primary keys should never change. Updating a primary key should always be out of the question. This is because it is most likely to be used in multiple indexes and used as a foreign key. Updating a single primary key could cause of ripple effect of changes.
3. Do NOT use "your problem primary key" as your logic model primary key. For example passport number, social security number, or employee contract number as these "primary key" can change for real world situations.

On surrogate vs natural key, I refer to the rules above. If the natural key is small and will never change it can be used as a primary key. If the natural key is large or likely to change I use surrogate keys. If there is no primary key I still make a surrogate key because experience shows you will always add tables to your schema and wish you'd put a primary key in place.

Natural verses artifical keys is a kind of religious debate among the database community - see this article and others it links to. I'm neither in favour of always having artifical keys, nor of never having them. I would decide on a case-by-case basis, for example:

• US States: I'd go for state_code ('TX' for Texas etc.), rather than state_id=1 for Texas
• Employees: I'd usually create an artifical employee_id, because it's hard to find anything else that works. SSN or equivalent may work, but there could be issues like a new joiner who hasn't supplied his/her SSN yet.
• Employee Salary History: (employee_id, start_date). I would not create an artifical employee_salary_history_id. What point would it serve (other than "foolish consistency")

Wherever artificial keys are used, you should always also declare unique constraints on the natural keys. For example, use state_id if you must, but then you'd better declare a unique constraint on state_code, otherwise you are sure to eventually end up with:

state_id    state_code   state_name
137         TX           Texas
...         ...          ...
249         TX           Texas

Just an extra comment on something that is often overlooked. Sometimes not using a surrogate key has benefits in the child tables. Let's say we have a design that allows you to run multiple companies within the one database (maybe it's a hosted solution, or whatever).

Let's say we have these tables and columns:

Company:
  CompanyId   (primary key)

CostCenter:
  CompanyId   (primary key, foreign key to Company)
  CostCentre  (primary key)

CostElement
  CompanyId   (primary key, foreign key to Company)
  CostElement (primary key)

Invoice:
  InvoiceId    (primary key)
  CompanyId    (primary key, in foreign key to CostCentre, in foreign key to CostElement)
  CostCentre   (in foreign key to CostCentre)
  CostElement  (in foreign key to CostElement)

In case that last bit doesn't make sense, Invoice.CompanyId is part of two foreign keys, one to the CostCentre table and one to the CostElement table. The primary key is (InvoiceId, CompanyId).

In this model, it's not possible to screw-up and reference a CostElement from one company and a CostCentre from another company. If a surrogate key was used on the CostElement and CostCentre tables, it would be.

The fewer chances to screw up, the better.