What is a custom object in Salesforce and What does a custom object permit the user to do?

Custom objects in Salesforce are the database tables that help users to collect the information that is unique to the organization.

Also note that with standard objects you are very limited in what you can change. You can add your own custom fields to them but you can't change the existing standard fields.  This is important to allow upgrades to the Salesforce platform (3 times per year) to be able to improve and add functionality to standard objects without breaking your system because you changed a standard object field.  Take care when creating custom objects - the temptation is always there to create a new custom object when you have a new requirement.  If you can adapt a standard object to suit your needs then always do this as long term it will be better for you. Read this article for more ... https://www.salesforce.com/blog/2016/10/you-should-use-standard-objects.html

Standard Objects -

They are the Objects which are already included in Salesforce. Examples - Accounts, Contacts, Cases etc.
They are common to every businesses.

Custom Objects -

They are created by companies or businesses to store information specific to their needs. Examples - Students, Properties etc.

This was very helpful.