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If a language is defined as:

$$ L = \{ w \in (a, b, c)^* \mid w = abc \} $$

Does this mean that in language $L$, a word consists of three symbols concatenated with each other? In other words, is '$a$' just one symbol (like '1' or '0') or can it be a longer string ('10..' '01..')?

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    If I wanted to imply that the alphabet of $L$ is $\{a,b,c\}$, I would write it as $L = \{ w \in \{a,b,c\}^* \mid w = abc\}$, but there's a good chance that the author means the same thing. Unless there's some other place where the alphabet is explicitly given. If indeed $\{a,b,c\}$ is the alphabet, then $a$ is, as you say, just one symbol. – 2017-01-30

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$(a,b,c)^*$ (typically written as $\{a,b,c\}^*$) is the set of all finite strings consisting of the symbols $a$, $b$, and $c$; see Wikipedia. Elements in $\{a,b,c\}^*$ are for example $a$, $abbb$, $abca$, and the empty string. Since your definition of $L$ requires $w = abc$, this language contains only a single element, namely $abc$.