Other answers have defined what dynamic SQL is, but I didn't see any other answers that attempted to describe why we sometimes need to use it. (My experience is SQL Server, but I think other products are generally similar in this respect.)
Dynamic SQL is useful when you are replacing parts of a query that can't be replaced using other methods.
For example, every time you call a query like:
SELECT OrderID, OrderDate, TotalPrice FROM Orders WHERE CustomerID = ??
you will be passing in a different value for CustomerID. This is the simplest case, and one that can by solved using a parameterized query, or a stored procedure that accepts a parameter, etc.
Generally speaking, dynamic SQL should be avoided in favor of parameterized queries, for performance and security reasons. (Although the performance difference probably varies quite a bit between vendors, and perhaps even between product versions, or even server configuration).
Other queries are possible to do using parameters, but might be simpler as dynamic SQL:
SELECT OrderID, OrderDate, TotalPrice FROM Orders
WHERE CustomerID IN (??,??,??)
If you always had 3 values, this is as easy as the first one. But what if this is a variable-length list? Its possible to do with parameters, but can be very difficult. How about:
SELECT OrderID, OrderDate, TotalPrice FROM Orders WHERE CustomerID = ??
ORDER BY ??
This can't be substituted directly, you can do it with a huge complicated CASE statement in the ORDER BY explicitly listing all possible fields, which may or may not be practical, depending on the number of fields available to sort by.
Finally, some queries simply CAN'T be done using any other method.
Let's say you have a bunch of Orders tables (not saying this is great design), but you might find yourself hoping you can do something like:
SELECT OrderID, OrderDate, TotalPrice FROM ?? WHERE CustomerID = ??
This can't be done using any other methods. In my environment, I frequently encounter queries like:
SELECT (programatically built list of fields)
FROM table1 INNER JOIN table2
(Optional INNER JOIN to table3)
WHERE (condition1)
AND (long list of other optional WHERE clauses)
Again, not saying that this is necessarily great design, but dynamic SQL is pretty much required for these types of queries.
Hope this helps.