Changing the maximum length of a varchar column?

I'm trying to update the length of a varchar column from 255 characters to 500 without losing the contents. I've dropped and re-created tables before but I've never been exposed to the alter statement which is what I believe I need to use to do this. I found the documentation here: ALTER TABLE (Transfact-SQL) however I can't make heads or tails of it.

I have the following so far (essentially nothing unfortunately):

alter table [progennet_dev].PROGEN.LE
alter column UR_VALUE_3

How do I approach this? Is there better documentation for this statement out there (I did some searches for an example statement but came up empty)?

You need

ALTER TABLE YourTable ALTER COLUMN YourColumn <<new_datatype>> [NULL | NOT NULL]

But remember to specify NOT NULL explicitly if desired.

ALTER TABLE YourTable ALTER COLUMN YourColumn VARCHAR (500) NOT NULL;

If you leave it unspecified as below...

ALTER TABLE YourTable ALTER COLUMN YourColumn VARCHAR (500);

Then the column will default to allowing nulls even if it was originally defined as NOT NULL. i.e. omitting the specification in an ALTER TABLE ... ALTER COLUMN is always treated as.

ALTER TABLE YourTable ALTER COLUMN YourColumn VARCHAR (500) NULL;

This behaviour is different from that used for new columns created with ALTER TABLE (or at CREATE TABLE time). There the default nullability depends on the ANSI_NULL_DFLT settings.

Increasing column size with ALTER will not lose any data:

alter table [progennet_dev].PROGEN.LE 
    alter column UR_VALUE_3 varchar(500) 

As @Martin points out, remember to explicitly specify NULL | NOT NULL

For MySQL or DBMSes other than MSSQL, you may need to use modify instead of alter for the column value:

ALTER TABLE `table name` 
modify COLUMN `column name` varchar("length");

ALTER TABLE TABLE_NAME MODIFY COLUMN_NAME VARCHAR(40);

I am using Oracle SQL Developer and @anonymous' answer was the closest, but kept receiving syntax errors until I edited the query to this. I changed alter to modify and there's no need to define column_name as column.