A mixing service can be used to mix one's funds with others', with the intention of confusing the trail back to the funds' original source. In traditional financial systems, the equivalent would be moving funds through banks located in countries with strict bank-secrecy laws, such as the Cayman Islands, the Bahamas and Panama.
Mixing helps protect privacy, but can also be used for money laundering - mixing illegally obtained funds. After laundering, the funds appear legitimate. Mixing large amounts of money may be illegal, being in violation of anti-structuring laws.
There has been at least one incident where a Bitcoin exchange has blacklisted "tainted" deposits descending from stolen bitcoins. Manual or lightly automated mixing methods can make detection of taint more difficult unless the exchange follows the trail, but this approach does protect privacy like a true mixing service would.
Caution: Mixing services may themselves be operating with anonymity. As such, if the mixing output fails to be delivered or access to funds is denied there is no recourse. Use at your own discretion.