Here's a list of orgs in BA that I know work with international volunteers or have worked with them in the past. Of course, your level of Spanish will influence who and how you can help out.

Contact the orgs both by email AND phone. And be persistent if they don't respond! It may not always mean that they don't want you--they might be too busy, might have too many volunteers, might not have a project. Call/email them again, or go to the site for clarification. That has been the best piece of advice from people I've spoken with. But of course, don't be too much of a pain :)

Does anyone have more orgs to add to this list? Or good advice?

Animal Rights

www.apema.org.ar

Arts and Culture

www.proyectarte.org
www.radiojai.com.ar
www.circosocialdelsur.org.ar
www.ibpsargentina.org.ar

Community Building

www.pilotosolidarios.org.ar
www.surnorte.org.ar
www.mediapilapais.com
www.acorn.org/index.php?id=10599
www.fpvs.org/home.php
www.cilsa.org/
www.responde.org.ar/

Environment

www.ecoclubes.org.ar
http://www.greenpeace.org/argentina/

Families/Youth

www.lifeargentina.org
www.leer.org.ar
www.familiasur.org
www.conviven.org.ar/
www.asociacionsud.org
www.aedin.org/flash/home.html
www.cimientos.org/voluntarios.php
http://www.fundacionreciduca.org.ar/
http://www.fundacionsendero.org.ar/ingles/ingles.html

Health

www.huesped.org.ar/

Human Rights

www.amnesty.org.ar/

Jewish focus

www.tzedaka.org.ar

LGBT

http://www.sigla.org.ar/

Politics

www.poderciudadano.org
www.democraciarepresentativa.org

I just saw this post on the CouchSurfing website (www.couchsurfing.com):

hello all,

i just met this women who has an incredible life story and runs a project that helps people in need of food or medical assistance. she absolutely needs medical Spanish speakers! i cannot do too much medically since my Spanish is still minimal, but i am going to start assisting her with food service or any other services needed while i work on my Spanish. if you cant get in touch by the website, i also have some direct contact information.
Matt

www.loscarasucias.org.ar/