Python; List index out of range [address book task]

Rather than think of addressbook as a two-dimensional array of information about people, think of each dimension separately. An addressbook is a one-dimensional array of persons. Each person is, in turn, a one-dimensional array of information.

Bringing that thinking to our code can make the purpose much clearer:

# An address boook is an arary of persons
addressbook = []

number = int(input("How many people would you like in your address book:"))

# Create several persons, adding each to addressbook as we go:
for _ in range(number):

    # Create one person:
    person = []
    person.append(str(input("\nPlease enter your full name:")))
    person.append(str(input("Please enter your home address:")))
    person.append(str(input("Please enter your hometown:")))
    person.append(str(input("Please enter your mobile number:")))

    # Add one person to addressbook
    addressbook.append(person)

# Now addressbook has several persons

# Display addressbook, thinking of it as two-dim array
for loop in range(number):
    print("\nName:",addressbook[loop][0],"\nHomeaddress:",addressbook[loop][1],"\nHometown:",addressbook[loop][2],"\nMobile number:",addressbook[loop][3])

# Display addressbook, thinking of it as array of persons:
for person in addressbook:
    print("\nName:",person[0],"\nHomeaddress:",person[1],"\nHometown:",person[2],"\nMobile number:",person[3])

Other notes:

The statement addressbook = ([]) is identical to addressbook = []. Parentheses in that context do absolutely nothing.

Multi-dimensional arrays list the row number first, the column second. That is to say, the first number gets you to a row, and the 2nd number gets you to an element of that row. More simply, say addressbook[loop][0], not addressbook[0][loop].

The expression range(0, number,1) is identical to range(number), and the latter is easier to read.

In Python, if you need to assign a value but don't care about it further, use _, like so: for _ in range(number):.

The expression str(input(...)) is identical to the expression input(...), since input returns an object of type str. I left those alone because I suppose that verbose style is required by your instructor.

When the time comes, learn about breaking long expressions into multiple lines. Many people find that a line longer than about 80 characters is difficult to read.

You prompt the user "Please enter your full name" (emphasis added) each time. Presumably their name hasn't changed since the previous iteration of the loop. You are actually asking for some third party's full name. Try "Please enter their full name" or "Please enter this person's full name" or some such.

Finally, realize that computer programs will be read more often than they are written. This is true for professionally-maintained programs with hundreds of contributing programmers and also of one-off programs written by you for your own benefit. Use any means necessary to communicate effectively with the reader(s) of your program--comments, clear variable names, whitespace, whatever it takes.