How do you implement one-to-one, one-to-many and many-to-many relationships while designing tables?

One-to-One relationship can be implemented as a single table and rarely as two tables with primary and foreign key relationships.
One-to-Many relationships are implemented by splitting the data into two tables with primary key and foreign key relationships.
Many-to-Many relationships are implemented using a junction table with the keys from both the tables forming the composite primary key of the junction table.

It will be a good idea to read up a database designing fundamentals text book.