The handheld tool with wire loops used to beat ingredients is a whisk.
A whisk usually has a handle attached to several flexible loops of metal wire. Moving it rapidly through a mixture blends ingredients smoothly and can incorporate air, a process called whisking or whipping. Cooks use whisks to beat eggs, mix batters, emulsify sauces, smooth gravies, and whip cream or egg whites into foams.
Different whisk designs suit different jobs. A balloon whisk has a broad, rounded head and is useful for aerating cream or egg whites. A flat whisk works well in shallow pans, especially when making roux or gravy. Spiral and gravy whisks are designed for sauces and liquids, while some modern whisks use individual metal balls instead of crossing loops.
Wire whisks are relatively modern, with their invention generally placed around 1840 and attributed to Thomas Dowding. Earlier cooks used bundles of twigs, sometimes choosing fruitwood that could add a subtle aroma. In Japan, the bamboo chasen has long been used to whisk matcha, but it is a specialized tea tool rather than the usual kitchen whisk described in this question.
A rotary egg beater performs a similar job mechanically, but it is a different utensil.