A soft search is an informal credit check. These types of checks are not visible to prospective lenders and only you can see them.
Soft searches usually happen when:
you check your own report on services like Pillar;
when your report is accessed for the purpose of an identity check;
or when a lender wants to show you your eligibility for a new financial product
A hard search leaves a record on your credit report, so prospective lenders looking at your credit report to decide whether or not to lend to you can see that you have applied for credit (and also whether you were accepted).
This usually occurs when you take out a financial product, like a credit card or a loan. This type of footprint will have an impact on your credit score.