RANKING Functions – ROW_NUMBER, RANK, DENSE_RANK, NTILE

ROW_NUMBER () OVER ([<partition_by_clause>] <order_by_clause>)
Returns the sequential number of a row within a partition of a result set, starting at 1 for the first row in each partition.

RANK () OVER ([<partition_by_clause>] <order_by_clause>)
Returns the rank of each row within the partition of a result set.

DENSE_RANK () OVER ([<partition_by_clause>] <order_by_clause>)
Returns the rank of rows within the partition of a result set, without any gaps in the ranking.

NTILE (integer_expression) OVER ([<partition_by_clause>] <order_by_clause>)
Distributes the rows in an ordered partition into a specified number of groups.

All the above definition and syntax are taken from BOL. It is difficult to explain above function anything else than what they are explained in BOL. Following example is excellent example from BOL again. This function explains usage of all the four function together in one query.
USE AdventureWorks;
GO
SELECT c.FirstName, c.LastName
,ROW_NUMBER() OVER (
ORDER BY a.PostalCode) AS 'Row Number'
,RANK() OVER (
ORDER BY a.PostalCode) AS 'Rank'
,DENSE_RANK() OVER (
ORDER BY a.PostalCode) AS 'Dense Rank'
,NTILE(4) OVER (
ORDER BY a.PostalCode) AS 'Quartile'
,s.SalesYTD, a.PostalCode
FROM Sales.SalesPerson s
INNER JOIN Person.Contact c
ON s.SalesPersonID = c.ContactID
INNER JOIN Person.Address a
ON a.AddressID = c.ContactID
WHERE TerritoryID IS NOT NULL
AND SalesYTD <> 0;