How to dynamically connect to multiple Databases at runtime by changing connection string in ASP.Net

Put the connection string in web.config say and then access it as following

    protected void ddlConnections_SelectedIndexChanged(object sender, EventArgs e)
    {
        string conString = "";
        if (ddlConnections.SelectedItem.Value == "2013")
        {
            conString = ConfigurationManager.ConnectionStrings["ConString_2013"].ConnectionString;
        }
        if (ddlConnections.SelectedItem.Value == "2014")
        {
            conString = ConfigurationManager.ConnectionStrings["ConString_2014"].ConnectionString;
        }
        SqlConnection con = new SqlConnection(conString);
        con.Open();
    }

protected void ddlConnections_SelectedIndexChanged(object sender, EventArgs e)
{
    string DB = "";
    if (ddlConnections.SelectedItem.Value == "2013")
    {
        DB = "2013"
    }
    if (ddlConnections.SelectedItem.Value == "2014")
    {
        DB = "2014"
    }
    string conString="Data Source=<YOUR SERVER>;Initial Catalog="+DB+";User ID=<USERNAME>;Password=<PASSWORD>";

    Session["conString"]=conString;
    SqlConnection con = new SqlConnection(conString);
    con.Open();
}

If you want use that connection in another page, simply add following to open connection

SqlConnection con = new SqlConnection(Session["conString"].ToString());
    con.Open();