You want to be an employee who receives a W-2.

Those who receive W-2's are legally employees. Those who receive 1099's are legally not employees but independent contractors. There are restrictions on who can legally be an independent contractor, but employers try to get around this. That is because the employer has to pay more taxes on an employee than an independent contractor.

If you are an independent contractor, you can't get unemployment if you lose your job (since you weren't an employee). You also pay higher taxes, since the employer pays half of the social security and medicare tax when you're an employee. As an independent contractor you pay all of those taxes.

You are an employee, not contract labor. They must withold taxes etc. from your pay.

There are also many other aspects of this, but this is usually the biggest concern.

They must pay unemployment insurance also

You generally must withhold federal income tax from your employees' wages. You withhold part of Social Security and Medicare taxes from your employees' wages and you pay a matching amount yourself. To figure how much to withhold from each wage payment, use the employee's Form W-4 and the methods described in Publication 15, Employer's Tax Guide and Publication 15-A, Employer's Supplemental Tax Guide (PDF).

Federal Unemployment (FUTA) Tax

You report and pay FUTA tax separately from Federal Income tax, and Social Security and Medicare taxes. You pay FUTA tax only from your own funds. Employees do not pay this tax or have it withheld from their pay. Refer to Publication 15, Employer's Tax Guide and Publication 15-A, Employer's Supplemental Tax Guide (PDF) for more information on FUTA tax.

They are telling you that they are hiring you and putting you on the payroll and taking out taxes. The 1099 is used for contractors and would mean that they were not taking out taxes on you, and you would have to pay (maybe) those taxes yourself later.

It means that they don't hire you as a 'private contractor', (1099).

They take responsibility for all required insurances and withholding,

and you have the benefits of local labor law, (overtime, vacation, insurances etc.),

that can be denied to 'private contractors'.