I'm developing a website where users will be able to sign in, and thus they need to be able to register.
Is it sufficient to include the registration functionality (or link to it) on the sign-in page? Should there be a link to create a new account on the home page? What if the site only uses third-party registration (e.g. Facebook or OpenID authentication)?
How do users expect to be able to sign up for an account? I'd like to make this as painless as possible.
Is it sufficient to include the registration functionality (or link to it) on the sign-in page?
Yes, it is perfectly fine to have a joint sign-in/registration form. Amazon & many online retailers work that way. Here's a good article how to design combo-forms.
Should there be a link to create a new account on the home page? What if the site only uses third-party registration (e.g. Facebook or OpenID authentication)?
Not necessarily. You can either have just Login link (like Stack Exchange) or a link called Login / Register (if you have the space). If you're offering multiple 3rd-party authentication systems then make a dedicated login page. Otherwise, just put one button Login with service.
How do users expect to be able to sign up for an account?
Users expect sign-up to be as painless as possible. They prefer it to be just email for username and a password. If there are some other details are necessary to improve their experience with the product, the need must be communicated clearly, e.g. tell them why you want their mobile phone number or home address.
What kind of website is it? Does it require a user to login for the core task, or most of the features are still accessible when the user is logged out?
Generally, the user won't signup to your website just like that. They require a reason. For the same, making signup/login a part of flow will much preferred. For example, on a site which lets a user comment on something and comment action requires user to be logged in, the trigger point could be the click event on comment button. Take example, http://imgur.com
While going in the flow, providing this signup/login form in a modal box will be a good option. Also when the user logs in, you can check the remember me checkbox by default, so that the user doesn't have to go through the same flow again.
If login/signup is required for the core task on your website, giving it on the front page first thing with reasons to signup would also not be a bad idea. Take http://tumblr.com for an example.
For any other use case, I expect to see login/signup button on top right corner of the page. Though, with proper highlighting it can be placed at other section of the page as well.
The question is phrased as 'Is it sufficient to...?' but surely the question should be 'How can I maximize conversion?'. Maximizing conversion comes as a direct result of making it as easy and painless as possible.
It's a minimal cost to have a Sign in and a Sign up link on the home page as well as a link on the Sign in page to sign up (and vice-versa), so why do any less.
Here's an example below from Groupon which does exactly this and in both cases of the Sign-in and sign-up form, they use facebook and regular sign up.
Note how they also cover themselves again for people who ignore the decision point at the top of the form and instead look immediately at the blue sign in or sign up button on each of the forms. They place an additional link to the right of the button - 'Not a member yet?' and 'Already a member?' which each links to the other version of the form.
Groupon are trying to maximize conversion (by making it very easy) for the user to switch at each decision point.
Once you can find one form, you can find the other - there's no chance of thinking - 'Oh! There's only a sign-in link - so where do I sign up then?'
