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Long before the idea of government welfare programs, taking care of the poor was firmly entrenched in Jewish communities. Giving money and food to those who could not afford the basic necessities, particularly around the holidays, is a standard practice that is rooted all the way back in the Biblical injunction for tithes. Continuing that tradition today, many local Tomchei Shabbos programs, funded and staffed by area volunteers, deliver food packages to needy families every week.
When the holiday of Purim was added by Mordechai and Esther, the mitzvos [obligations] of the day were set to include recounting the story of the Book of Esther by hearing the Megillah (both at night and during the day), sending mishloach manos to friends, having a celebratory meal, and the mitzvah ofmatanos l'evyonim, that is gifts to the poor.
It is not only unseemly to indulge in making merry while the poor go hungry, it is absolutely contrary to the halacha [Jewish law]. The minimum prescribed by the halacha is to give one mishloach manos and two matanos l'evyonim, which indicates a direction for priorities. If people are compelled to curtail their holiday spending due to their own budget constraints can minimize their own feasting and give the minimal mishloach manos but should not hold back from giving to the poor.
Ideally, money is to be distributed on the day of Purim itself. In Jerusalem, where Purim is celebrated on the 15th of Adar, rather than the 14th, on the day of Shushan Purim, the holiday can fall out on the Sabbath. (Because of how the calendar is set, the 14th of Adar never falls out on the Sabbath.) In that case, the matanos l'evyonim are distributed a day early rather than late.
If you are unsure of where to give, area synagogues and schools often set up collections. A number of charities, like Yad Eliezer (which has an excellent rating from Charity Navigator) accept online credit card donations and can guarantee distribution on Purim.