Three examples that are not 1NF and a solution:

1: This is not 1NF because you have multiple email addresses stored in one field. Non-atomic data stored in a field.
user (id*, name, email)
    id    name    email
    1    jim        jim@jimmartinson.com,jim.martinson@ridgewater.edu
    
2: This is not 1NF because you have repeating fields used to store multiple email addresses. This makes queries complex and allows for redundant data.
user (id* name, email1, email3)
    id    name    email1                    email2                            email3 ( Need to modify the database to add a 3rd email address )
    1     jim     jim@jimmartinson.com      jim.martinson@ridgewater.edu
    
3: This is not 1NF because you have repeating tables used to store multiple email addresses. This makes queries complex and allows for redundant data.
user (id*, name)
    id   name
    1    jim
email1 (id*, email)
    id   email
    1    jim@jimmartinson.com
email2 (id*, email)
    id   email
    1    jim.martinson@ridgewater.edu
email3 ( Need to modify the database to add a 3rd email address )
    id   email

solution: Move email address to its own table and setup a one-to-many relationship. 
user (id*, name)
    id   name
    1    jim
useremail (id*@, email*, primary)
    id   email                           primary
    1    jim@jimmartinson.com            0
    1    jim.martinson@ridgewater.edu    1


Symbol key:
* = primary key
@ = foreign key
^ = unique field


An example from Database-Design-2nd-Edition:
UNF:
Student_Grade_Report (StudentNo, StudentName, Major, CourseNo, CourseName, InstructorNo, InstructorName, InstructorLocation, Grade)
StudentNo,  StudentName,  Major,  CourseNo,  CourseName,           InstructorNo,  InstructorName,  InstructorLocation,  Grade
1           jim           CP      CST1600    Database Design       1              jim              327                  NULL
1           jim           CST     CST1500    Hardware Diagnostics  1              jim              327                  NULL

1NF ( I found UNF to be in !NF and consider this a 2NF first draft):
Student (StudentNo*, StudentName, Major)
StudentNo*,  StudentName,  Major
1            jim           CP

StudentCourse (StudentNo*@, CourseNo*, CourseName, InstructorNo, InstructorName, InstructorLocation, Grade)
StudentNo*@,  CourseNo*,  CourseName,           InstructorNo,     InstructorName,  InstructorLocation,  Grade
1             CST1600     Database Design       1                 jim              327                  NULL
1             CST1500     Hardware Diagnostics  1                 jim              327                  NULL


2NF:
Student (StudentNo*, StudentName, Major)
StudentNo*,  StudentName,  Major
1            jim           CP

CourseGrade (StudentNo*@, CourseNo*, Grade)
StudentNo*@,  CourseNo*,  Grade
1             CST1600     NULL
1             CST1500     NULL

CourseInstructor (CourseNo*, CourseName, InstructorNo, InstructorName, InstructorLocation)
CourseNo*,  CourseName,           InstructorNo,  InstructorName,  InstructorLocation
CST1600     Database Design       1              jim              327
CST1500     Hardware Diagnostics  1              jim              327


3NF:
Student (StudentNo*, StudentName, Major)
StudentNo*,  StudentName,  Major
1            jim           CP

CourseGrade (StudentNo*@, CourseNo*, Grade)
StudentNo*@,  CourseNo*,  Grade
1             CST1600     NULL
1             CST1500     NULL

Course (CourseNo*, CourseName, InstructorNo)
CourseNo*,  CourseName,           InstructorNo
CST1600     Database Design       1
CST1500     Hardware Diagnostics  1

Instructor (InstructorNo, InstructorName, InstructorLocation)
InstructorNo,     InstructorName, InstructorLocation
1                jim                327


I usually look at the data and look for entities (tables). What are the real-world things I am keeting data about?
Student_Grade_Report (StudentNo, StudentName, Major, CourseNo, CourseName, InstructorNo, InstructorName, InstructorLocation, Grade)
    student
    course
    instructor
    student_course