Income tax returns can be difficult. Breaking the 1040 down into sections helped me to understand it better.
Why we file taxes on April 15th
We need to balance our "tax checkbook" once a year. April 15th has been coined "Tax Day" as this is the deadline that all taxpaying Americans must submit their balanced tax checkbook to the IRS.

This allows employers to mail tax withholding and earning statements (better known as W-2s) and the self-employed the time necessary to get their books in order.
Form 1040: Two pages to zero
When you boil it down, there is only one form that reports your income tax: Form 1040. This two page document collects all your personal data into one place and it tells you whether you get a refund or owe more in taxes.
Eight sections to refund
There are eight sections that must be completed before calculating a refund:
Information
For identification and filing purposes only. This has no bearing on your taxes.
Filing Status
You only need to identify how you will file and then check the appropriate box.
Exemptions
The exemption section will have a huge impact on the second page. List yourself and all your dependents here and you will see your taxable income go down on Page 2.
Income
This is the place to be! List all the sources of money that came in during the year like

Wages
Tips
Interest and investment income
Social security and Pensions
Rental income
Annuities (remember, these are taxed at ORDINARY INCOME RATES). Think about it

Line 22 is the total of all these sources of money. It is your Gross Income. Take a good look at it, you might get a wake-up later down the form.
Adjustments to income 
You can reduce the amount of your gross income with qualified

Student Loan interest paid
Educator expenses (back for 2012!)
IRA deductions (similar to the effect of putting money into a 401k)
Health Savings Account contributions (HSA)

While the first one, Student Loan Interest, isn't good for you, it is good to report on your 1040. The others are wonderful tax saving vehicles!
Tax and Credits
This section gets a bit wonky. It performed three actions: It reduces taxable income, finds what your total tax liability is, and then reduces the tax liability with non-refundable credits.
Should I take the Standard Deduction or the Itemized Deduction?
Nearly everyone that files a tax return gets to reduce their taxable income by at least the Standard Deduction. A married couple with an Adjusted Gross Income (AGI) of $100,000 filing their tax return jointly automatically will get to reduce their liability by $11,900.

However, if they have qualified expenses and charitable giving that exceeds that amount it would make more sense to Itemize. Example:
A married couple pays $5,000 in mortgage interest, $3,000 in real estate taxes, and gave $5,000 to their church (a qualified 501c3). This allows them to reduce their liability by $13,000 instead of the standard deduction of $11,900.

Exemptions
Here is where it pays to have children (in a tax-benefit sort of way). Each qualified person in the Exemption section from Page 1 of the Form 1040 gets to reduce the tax liability even more by $3,800 each in 2012.
The family of four with an AGI of $100,000 reduce the amount that on which their income tax is calculated from the Itemized Deduction ($13,000 - bringing the number down to $87,000) and with personal Exemptions (another $15,200. This brings the amount of income they calculate their tax on to $71,800.

Wait, there's more tax adjustments!
The tax on $71,800 turns out to be about $8,000. But the Tax and Credits section also allows for non-refundable credits like

Retirement Savings Contributions Credit
Child Tax Credit
Dependent Care Credit

These don't reduce the taxable income, THEY REDUCE YOUR TAX! That's pretty powerful stuff!
Other Taxes
Your guess is as good as mine. We have over 73,000 pages of our tax code to refer to.