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LingakuLingakuてくれる says someone does something for you, or for someone on your side – your family, your friends, your group. It goes after the て-form of a verb, and the person doing the favour is the subject.
Without くれる the sentence only reports what happened. たなか田中さんがおしえた教えた says Tanaka taught; たなか田中さんがおしえて教えてくれた says Tanaka taught you, and you are glad of it. Leaving くれる out when you were the one helped sounds cold.
The favour in てくれる always comes towards you. When you do something for someone else, it is てあげる.
The person who benefits is usually left unsaid. When you do name them, に only works with verbs that already take a person with に, such as おしえる教える or かす貸す. Otherwise the person goes on the thing with の.
Both say you received a favour. With てくれる the giver is the subject and takes が. With てもらう you are the subject, and the giver takes に.
Among friends, てくれる？ asks someone to do something for you, and てくれない？ is a little softer.
When the giver is above you, the honorific てくださる replaces てくれる. てください, "please do", is its request form.