Lewis structures are a simple way of drawing molecules to show the bonding and "location" of electrons. Despite their simplicity, it turns out that they are very useful in predicting 3-dimensional structure (Through VSEPR theory.) and some aspects of reactivity.
The American chemist Gilbert Lewis suggested in 1916 that since noble gas atoms are very stable, non-noble gas atoms can share electrons with each other to gain noble gas structure. Atoms sharing two electrons are said to form a covalent bond between the two atoms.
For example, a hydrogen atom has a single electron. The noble gas helium has two electrons. By sharing an electron with another hydrogen atom, the two can form a covalent bond.

Now each hydrogen atom is surrounded by two electrons, and has the same electronic structure as helium.
In a Lewis structure, individual electrons are shown as dots and covalent bonds as lines connecting the symbols for the elements. (Multiple lines for multiple bonds.) Only the valence electrons are shown. For example, the Lewis diagram for the HF molecule is shown below:

There is a bond between the hydrogen and fluorine atoms: this is the single line. There are six electrons in lone pairs around the fluorine atom. The hydrogen atom has two electrons and thus the same electronic structure as helium, the fluorine atom has eight total electrons (two from the bond, six in lone pairs.) which is the same as the noble gas neon.
Some examples of Lewis structures are shown below. Note how each atom has noble gas structure for the valence electrons

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