The difference between claiming 0 and 1 on a tax return is that 0 means the taxpayer claims no exemptions while 1 means the taxpayer claims one exemption, according to the IRS. A taxpayer may take one exemption for each person for whom he is financially responsible.
The IRS notes that each exemption reduces a taxpayer's taxable income by $3,650, as of 2014. A single person who is financially responsible for just himself claims one exemption. A person who is claimed as a dependent on another person's tax return, such as a child, must claim zero exemptions. A person who is financially responsible for himself and others may claim one exemption, plus an additional exemption for each of his dependents.
When employees fill out W-4 forms, they should be aware of how many exemptions they are claiming on their tax return for that year, according to Rapid Tax. The number of exemptions the employee claims is used to determine how many allowances need to be claimed on a W-4 form. For example, a person declared a dependent on another taxpayer's return would claim zero allowances, while a person who is single and financially responsible for himself only would claim one allowance.