Let $X$ be an irreducible algebraic variety over some field $k$. It is well known, that if $X$ is smooth and of dimension $d$, then the tangent bundle of $X$ is smooth, irreducible and of dimension $2d$. There is no difficulty in defining the tangent bundle $T_X$ in the non-smooth case. It is defined as the (global) Spec of the symmetric algebra of the sheaf of Kähler differentials $\Omega^1_{X|k}$. In fact this definition works for an arbitrary morphism of schemes $X\to S$. (See EGA IV$_4$ 16.5.12.1.)

All the irreducible components of $T_X$ come from points of $X$. See Prop. 2.2 in this paper for a precise statement. The generic point of $X$ always yields an irreducible component of $T_X$ of dimension $2d$, but typically there will be other components.

I would like to know if there is an example such that $T_X$ has an irreducible component of dimension strictly smaller than $2d$.

Note that this will not be possible if $X\subset\mathbb{A}^n_k$ is a complete intersection: If the ideal of $X$ is generated by $f_1,\ldots,f_{n-d}$ then $T_X$ is described by the $2(n-d)$ equations $f_1,\ldots,f_{n-d},d(f_1),\ldots,d(f_{n-d})$ in $\mathbb{A}^{2n}_k$. (Cf. Example 1.5 in the above paper.)