Both shiru (知る) and wakaru (わかる) have similar meanings. Shiru is usually used in the te iru form (see How does the te form work?), shitte iru, to talk about knowing something.
In some cases, either wakaru or shiru can be used:
Tsukaikata ga wakaranai node oshiete kudasai (使い方がわからないので教えてください)
Tsukaikata wo shiranai node oshiete kudasai (使い方を知らないので教えてください)"I don't know how to use it, so please tell me". However, in some cases shiru cannot be used:
Wrong: Iroiro tameshite mita ga tsukaikata wo shiranai (×いろいろ試してみたが使い方を知らない)
OK: Iroiro tameshite mita ga tsukaikata ga wakaranai (いろいろ試してみたが使い方がわからない)"I tried several things, but I don't understand how to use it".
If we want to ask about knowing a person, "Do you know him?",
Kare wo shitte imasu ka?is OK, but
Kare wa wakarimasu ka?sounds more like "Do you understand him?"
There are also some differences in usage and syntax. For example, shiritai (知りたい) "want to know" is possible, but wakaritai (わかりたい) is not, and the passive form shirarete iru (知られている) "is known about" is possible, but wakararete iru (わかられている) is not.
The negative, shiranai, has a special meaning of disclaiming responsibility. It also carries the implication that the speaker does not care about the answer, so it might be considered rude to answer a teacher's question with shirimasen rather than wakarimasen. In the form shiranpuri (知らんぷり), it means "pretending not to know" or ignoring someone.
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