In theory, all fruits and vegetables are kosher, with no certification needed. Still, there are two areas of concern.
The first is the problem of insect infestation. Some kinds of produce, like leafy green vegetables, broccoli, and strawberries are often infested with insects, which aren't kosher.
For this reason, produce must be checked carefully before eating. Star-K Kosher Certification has posted several videos showing how to check for insects.
The second problem is a bit obscure, and concerns Israeli produce. Jewish farmers in Israel are required to offer tithes from their produce. Additionally, in Sabbatical years they are required to let their land lie fallow. For these reasons, Israeli produce requires kosher certification. The kosher supervisors insure that the proper tithes are given, and that in Sabbatical years, only produce from gentile-owned farms can be certified as kosher.
