I am providing part-time childcare for a friend's child. My friend is able to be reimbursed for her childcare expenses by the company she works for if she gives them my name, social security number and weekly receipts for the childcare expenses. If I give her company the information they are requesting so she can be reimbursed, will I then need to declare my childcare income and pay taxes on it? If so, will I need to be paying taxes as a business or just as it were extra income - I'm wondering if this would really reduce the amount of money I will be earning, as I am not hardly charging her in the first place.

Even if you don't give your name and ss to the company, you must still declare this income. This is self employment income. You must file tax return if yourself employment income is $400 or more.
You can deduct your business related expenses from the income. On the net income (income minus expenses) you will be paying SE tax at 15.3%. Then you will have federal income tax and state income tax depending upon your total income.

There are many, many deductions that you may use to reduce your taxable income. The biggest one is the percentage usage of your home as a day care. I would recommend going to a qualified tax preparer for your first year of filing a tax return as a day care provider. Make sure that they have experience with preparing a day care provider schedule C as there are a lot of deductions that an inexperienced preparer may overlook.

I believe it all depends on how much you actually earn. If it's more than $400 then legally you have to declare it. If this particular child is the only one you provide daycare services for then you should be able to declare it as just extra income and not as a "business."

Form 1040 has line 21 is for "Other income". The instructions of line 21 clearly state that you can't put self employment income on this line. And even if you don't have any business expenses, still use schedule C or C-EZ (Form 1040).
There is no way you can avoid paying SE taxes on this income. You income is self employment (business income).
This income is treated as Earned Income so you will get Earned Income Credit if your income is below a certain level and you are over 25. And if you have a qualifying child, you can even get Child Tax Credit/Additional Child Tax Credit, which can result in getting more than what you will actually pay.

All the advice given above is valid, with one exception.

The AGE 25 rules applies only if you are single and have NO children of your own.

If you have one or more children, you qualify for the Earned Income Credit regardless of you age.