Normal DELETE :
DELETE
FROM Table
WHERE column1 = 'some value'
DELETE WITH INDEX :
DELETE Table1
FROM Table1 WITH(INDEX(PK_Table1))
WHERE column1 = 'some value'
I know that when you delete a row from a table it will search for all reference tables, which slows down delete execution.
How does delete using WITH INDEX work?
WITH(INDEX(PK_Table1))
is an index hint; i.e. tells SQL to use a specific index rather than trying to determine the best one. Generally index hints aren't needed; SQL is very good at knowing what's best; and doesn't take much time working out the best index, since it recalls which index it used the last time it had a similar query.
There are some cases where SQL may get it wrong; for those you could improve performance by adding a hint. These cases are very rare. In such cases, test thoroughly, and with data as close to real-world as possible (as what data's in the table will impact how well the query performs under each index
Unless you're seeing a real performance problem & can prove that your hint helps, don't use the hint.
Here's a related post on hints, which summarises the argument well: https://www.brentozar.com/archive/2013/10/index-hints-helpful-or-harmful/