Difference between Primary Key and Unique Index

Primary key is a combination of unique+not null+cluster 
index and it will not allow any duplicate values, null 
values, and it's data will be arranged in ascending order 
due to clustered index.  Only one primary key is allowed 
per table.

Unique key is a key which will not accept any duplicate 
values, but it will accept one null value.  We can place 
any number of unique keys per table.
 
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A primary key is one which is used to identify the row in question. It might also have some meaning beyond that (if there was already a piece of "real" data that could serve) or it may be purely an implementation artefact (most IDENTITY columns, and equivalent auto-incremented values on other database systems).
A unique key is a more general case, where a key cannot have repeated values. In most cases people cannot have the same social security numbers in relation to the same jurisdiction (an international case could differ). Hence if we were storing social security numbers, then we would want to model them as unique, as any case of them matching an existing number is clearly wrong. Usernames generally must be unique also, so here's another case. External identifiers (identifiers used by another system, standard or protocol) tend to also be unique, e.g. there is only one language that has a given ISO 639 code, so if we were storing ISO 639 codes we would model that as unique.
This uniqueness can also be across more than one column. For example, in most hierarchical categorisation systems (e.g. a folder structure) no item can have both the same parent item and the same name, though there could be other items with the same parent and different names, and others with the same name and different parents. This multi-column capability is also present on primary keys.
A table may also have more than one unique key. E.g. a user may have both an id number and a username, and both will need to be unique.
Any non-nullable unique key can therefore serve as a primary key. Sometimes primary keys that come from the innate data being modelled are referred to as "natural primary keys", because they are a "natural" part of the data, rather than just an implementation artefact. The decision as to which to use depends on a few things:
  1. Likelihood of change of specification. If we modelled a social security number as unique and then had to adapt to allow for multiple jurisdictions where two or more use a similar enough numbering system to allow for collisions, we likely need just remove the uniqueness constraint (other changes may be needed). If it was our primary key, we now also need to use a new primary key, and change any table that was using that primary key as part of a relationship, and any query that joined on it.
  2. Speed of look-up. Key efficiency can be important, as they are used in many WHERE clauses and (more often) in many JOINs. With JOINS in particular, speed of lookup can be vital. The impact will depend on implementation details, and different databases vary according to how they will handle different datatypes (I would have few qualms from a performance perspective in using a large piece of text as a primary key in Postgres where I could specify the use of hash joins, but I'd be very hesitant to do so in SQLServer [Edit: for "large" I'm thinking of perhaps the size of a username, not something the size of the entire Norse Eddas!]).
  3. Frequency of the key being the only interesting data. For example, with a table of languages, and a table of pieces of comments in that language, very often the only reason I would want to join on the language table when dealing with the comments table is either to obtain the language code or to restrict a query to those with a particular language code. Other information about the language is likely to be much more rarely used. In this case while joining on the code is likely to be less efficient than joining on a numeric id set from an IDENTITY column, having the code as the primary key - and hence as what is stored in the foreign key column on the comments table - will remove the need for any JOIN at all, with a considerable efficiency gain. More often though I want more information from the relevant tables than that, so making the JOIN more efficient is more important.

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