My mother in law visited me last year and she stayed with me for six months and she doesn't have a social security number. Can I claim her as a dependent in my tax return?

It depends. First - where is your mother from? A dependent must be a US citizen, US resident alien, US national, or resident of the Canada or Mexico. Since she does not have a SS number I'm assuming she is not a US resident, and so she must be from either Canada or Mexico. Further, you must have provided over half her support for the year, and her income must be less than $3650. If she meets all these criteria then you may treat her as a dependent. She will need to apply for an Individual Taxpyaer Identification umber (ITIN), using for W-7. See: http://www.irs.gov/pub/irs-pdf/fw7.pdf