Algorithm to check the 2∀-connectness property of a graph
A simple algorithm is the following:

Let $G = (V,E)$ be the graph you want to check for 2-connectivity.


  For each edge $e = (x,y) \in E$, look if there is a path from $x$ to $y$
  in $(V, E \setminus \{e\})$. If for some $e$, there is not such a path, the graph is
  not 2-edge connected. If there is such a path for each $e$, $G$ is
  2-edge connected.


You should also note, that this algorithm is not the fastest, but very easy to understand. The path-finding part can be done using for example BFS or Dijkstra's Algorithm.

This article also shows a simple algorithm for testing 2-edge connectivity, which runs in linear time.