I only worked for 3 months this year at a job that paid about $2000 a month, and made about $1500 a month for 5 months. Can my husband claim me as a dependent?

No your husband cannot claim you as a dependent.   A spouse is never claimed as a dependent.   File a joint return and enter ALL of your combined income on that return.

If you were legally married at the end of 2021 your filing choices are married filing jointly or married filing separately.

Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will get the married filing jointly standard deduction of $25,100 (+$1350 for each spouse 65 or older)  You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit. 

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return. Some of the special rules for filing separately include: you cannot get earned income credit, education credits, adoption credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. Your limit for SALT (state and local taxes and sales tax) will be only $5000 per spouse. In many cases you will not be able to take the child and dependent care credit. The amount you can contribute to a retirement account will be affected. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. ( Community property states:  AZ, CA, ID, LA, NV, NM, TX, WA, WI)

 If  you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice.

https://ttlc.intuit.com/questions/1894449-married-filing-jointly-vs-married-filing-separately

https://ttlc.intuit.com/questions/1901162-married-filing-separately-in-community-property-states

https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separately

If your spouse has debts that might cause your refund to be seized, you can file as "injured spouse" to protect the part of the refund that can be attributed to your earnings.   Since you did not earn much, it might not help a whole lot.

INJURED SPOUSE

https://ttlc.intuit.com/questions/1910698-how-do-i-file-form-8379-injured-spouse-allocation

 https://www.irs.gov/uac/tax-refund-offsets-pay-unpaid-debts

https://ttlc.intuit.com/community/refunds/help/what-is-a-refund-offset/00/26301

Reduced Refund      https://www.irs.gov/taxtopics/tc203

 NOTE: You can contact the IRS Treasury Offset Program Call Center at 1-800-304-3107 to ask if they have an offset for you on file. TurboTax would not have that information.

https://turbotax.intuit.com/tax-tips/tax-payments/who-can-garnish-an-income-tax-refund/L7cPPzDyc

Sorry if you are married your never can claim a spouse as a dependent.   You just file a Joint return.  That's about the same as claiming a dependent.  Actually better since you get a higher Standard Deduction for Joint.  

You might need to file an Injured Spouse form to protect your refund.   FAQ Injured Spouse Form 8379

https://ttlc.intuit.com/community/accessing/help/how-do-i-file-form-8379-injured-spouse-allocation/01/26228

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@jdevine1996 wrote:
And who should claim our child if we live together and both cared for him equally throughout the year?

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If you file as married filing jointly, the two of you together claim your child as a dependent on your joint tax return. It's not one or the other of you claiming the child, it's both of you together.

If you and your spouse file married filing separately, then either one can claim the child (but not both).  As long as the child is either the biological or legal step-child of both of you, it doesn't matter who claims the child.  You will have to test both scenarios to know what works best for you.]

If you file separately, you will protect any refund you claim on your return, without the need for an injured spouse form.  But you will likely pay more overall tax.  Certain tax benefits are reduced or disallowed when married filing jointly.  Also, if either spouse itemizes their deductions, both spouses must itemize, even if one spouse has no deductions (they take zero deduction).  You can't stack all the itemized deductions on one spouse and the other spouse take the standard deduction. 

As far as the injured spouse form is concerned, that can only be used to ask the IRS to keep your refund separate from your spouse's debts incurred before the marriage.  Any debts during the marriage when you signed a joint return are equally the responsibility of yourself and your spouse.

Also, your federal tax refund can only be taken for federal debts, student loans, child support, and taxes or repayments related to unemployment compensation.  Any other debts might be able to attach a state tax refund, but can't attach a federal tax refund.  So you may be at less risk than you think. 

Q. Can my husband claim me as a dependent?

A.  No.

Or more precisely: technically no.  You are allowed to file a joint return with your spouse, regardless of how much income you have, even if your income is zero.  Filing as Married Filing Jointly gets you all the benefits of being a dependent and much more.