See the Wikipedia article on lift:
In a category $C$, let be given two arrows $f,g$ with codomain $Z$. A lift (of $f$ with respect to $g$) is an arrow $F$ such that $g\circ F =f$. So it is a morphism from $f$ to $g$ in the over category $C/Z$.
Motivation: the idea is that $g:X\to Z$ indicates the vertical direction pointing down: $X$ lies above $Z$ via $g$. Think of $g$ as a fibre bundle, or covering map, or vertical projction. So a given $f:Y\to Z$ takes values in the lower space; and to lift $f$ (via $g$) means to 'let it take values in the upper space': find a 'modified' $F:Y\to X$ such that, after vertically projecting its values, we get $f$ back; in other words $p\circ F=f$. If we take $Y=[0,1]$ this is path-lifting.
- a lift of the identity $id:Z\to Z$ (via $g:X\to Z$) is an arrow $s:Z\to X$ such that $g\circ s=id$, i.e. a section of $g$.
(I believe in your case this is meant: a section of a quotient map $q:X\to X/\sim$ is a map $s:X/\sim\to X$ such that $s[x]\sim x$. One could say that $s[x]\in X$ is a lift of $[x]$.)
DUALLY: let be given two arrows $f,g$ with domain $Z$. A lift of $f$ (via $g$) is an arrow $F$ such that $F\circ f=g$. So it is a morphism from $f$ to $g$ in the under category $Z/C$.
Motivation: again the idea is that $g:Z\to X$ indicates the vertical direction, but now pointing up: $X$ lies above $Z$ via $g$. Think of $g$ as an
- a lift of $f:Y\to X$ (via an inclusion $g:Z\to X$) is an arrow $F:Y\to X$ such that $F\circ i=g$, i.e. an extension of $f$.
Related concepts are injective and projective objects. Moreover note that lifts need not exist in general (see also Zhen Lin's comment).