El Poblado is the preferred place to stay for most travelers. It's close to the bars and restaurants, and is usually safe, even late at night.
Those who want a less-sanitized experience of Medellín may prefer the more rough-and-tumble center, although you'll want to take taxis after dark.
A middle option is the area around Laureles/La Setenta, which is less flashy than Poblado yet more orderly than the center.
El Poblado has several different sub districts with different kinds of accommodations. Most hostels in the blocks around Parque Lleras tend to be ‘party hostels’ so bring ear plugs. The Patio Bonito area close to the metro has more relaxed accommodations and a quieter vibe. Another option is Manila, where there are several good hostels dotted among hip restaurants and bars.