In ASP.NET; you have a Session cookie. This cookie is used to identify which session is yours; but doesn’t actually contain the session information.
By default, ASP.NET will store session information in memory inside of the worker process (InProc), typically w3wp.exe. There are other modes for storing session, such as Out of Proc and a SQL Server.
ASP.NET by default uses a cookie; but can be configured to be “cookieless” if you really need it; which instead stores your Session ID in the URL itself. This typically has several disadvantages; such as maintence of links become difficult, people bookmarking URLs with expired session IDs (so you need to handle expired session IDs, etc). Most modern phones, even non-smart phones, support cookies. Older phones may not. Whether you need to support cookieless sessions is up to you.
If your URL looked like this:
http://www.example.com/page.aspx
A cookieless URL would look like this:
http://www.example.com/(S(lit3py55t21z5v55vlm25s55))/page.aspx
Where lit3py55t21z5v55vlm25s55 is a session ID.
You can learn more about ASP.NET’s session state here