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The irregularity of a graph $G=(V,E)$ is defined as the sum of imbalances $|d_u-d_v|$ over all edges $uv\in E$, where $d_u$ denotes the degree of the vertex $u$ in $G$. This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with $n$ vertices and $p$ pendant vertices ($1\le p\le n-1$), and characterize the corresponding extremal graphs.