Shaatnez is the occurrence of wool and linen in the same garment. There are various prohibitions (Lev 19:19, Deut 22:911) against the mingling of different kinds; this is one of them. A linen tie worn with a wool suit is permitted, but a wool suit with linen threaded buttons is prohibited.
While in practice, many garments do not have any shaatnez and may be assumed to have none, the particulars vary by garment type. The padding in many garments such as suits or the embroidery thread, such as designs on sweaters (men's and women's) may cause shaatnez problems. The padding filler in many suits is made of assorted rags which may be mixed linen and wool in themselves (so it is not just a worry of linen threaded padding in a wool shell suit).
Today, the usual way of observing the shaatnez prohibitions is to first check the fabric list (careful: lana/lino is Spanish for wool/linen). If the fabric list shows a forbidden mixture, don't bother, you probably can't get it fixed. If the label indicates "other" it may or may not be linen. Even if the label says the garment is 100% wool, there may still be problems.
Since the fabric list on suits usually refers only to the shell (and ignores padding or ornamental threads), the label can only be used to identify garments that definitely have shaatnez. Thus, if the label indicates that the suit (for example) can be good, take it to a shaatnez lab for testing. Most cities with at least a medium sized Orthodox community have qualified shaatnez testers. If the city has a local Vaad Hakashrus they can usually refer you to a reliable tester.