The Lasso is the solution to
$$\hat{\beta} \in argmin_\beta \frac{1}{2n}\Vert y-X\beta\Vert_2^2 + \lambda\Vert\beta\Vert_1$$
Evidently, $\hat{\beta}$ also depends on $\lambda$, so really, we could write this dependence explicitly: $\hat{\beta}(\lambda)$. Thus, we can interpret $\hat{\beta}$ as a function $\lambda\mapsto\hat{\beta}(\lambda)$, whose domain is $[0,\infty)$. The "solution path" of the Lasso is precisely this function: For each $\lambda\in[0,\infty)$, there is a vector $\hat{\beta}(\lambda)$ that solves the optimization problem above. When $\lambda=0$, you start out at the OLS solution, and as you increase $\lambda$, the Lasso solution changes to become more and more sparse, until ultimately $\lambda=+\infty$ and $\hat{\beta}(+\infty) = 0$.
Many approximate algorithms for the Lasso solve this optimization problem at some discrete values $\lambda_1,\ldots,\lambda_L$. This is simple and efficient, but it does not provide the full spectrum of possible solutions, since there are infinitely many possible choices of $\lambda$.
LARS, and other related algorithms, on the other hand, provide the full solution path for all possible $\lambda$ (i.e. infinitely many solutions). This why they are commonly referred to as "path algorithms" since they provide a parametrized solution "path" for all $\lambda$.
Note: To understand the use of the word "path", think of parametric equations and lines from calculus e.g. $f(t) = (x(t), y(t))$. As $t$ varies, $f(t)$ plots a path in space. For the Lasso, replace $t$ with $\lambda$ and we have $\hat{\beta}(\lambda) = (\hat{\beta}_1(\lambda),\ldots,\hat{\beta}_p(\lambda))$.
