Unregistered groups can't, of course. Not having tax-deductible status may not hurt donations, however. Many individuals happily contribute to causes they care for without concern for a tax deduction. Even when they can, many don't claim their deductions. Corporations can take their tax deductions without official receipts, as explained below.
Foundations can only give to registered charities, however.
Gifts of money are tax deductible, whether made in cash, by cheques, via credit card or money order. Pledges are not deductible until they are paid. Gifts of goods are tax deductible. Gifts of services are not deductible. It is legal to exchange cheques to solve this problem.