Quick version:
Use a web service midlayer running on a public host you control (possibly but not necessarily the database host). Expose public web service methods to do the limited work you want to permit and nothing else.
Related questions:
Implementation options
Personally I'd use a Java application server like Apache Tomcat or JBoss AS 7 and I'd write my web service methods using JAX-RS to produce a nice REST-style API for my app to use. That's what I'm familiar with and it works well, but you have lots of options including implementations of:
REST-like APIs (Java's JAX-RS impls Jersey and RESTEasy, various other langs tools) that use HTTP requests and produce JSON or XML replies.
SOAP with WSDL, the classic "web service" layer. In Java done with JAX-WS among other options. Most languages have tools for SOAP+WSDL but it's kind of crappy to work with especially on intermittently connected devices like mobiles.
XML-RPC if you like pain
There are some JAX-RS quickstarts on the JBoss AS 7 quickstarts list; just search for "JAX-RS". The "kitchen sink" quickstart is useful, though perhaps not ideal if you're not familiar with the basics of JBoss AS 7 and Jave EE 6. Fort the JAX-RS specifics you're better off with a Jersey or RESTEasy tutorial like this or this.
Important considerations
Use HTTPs if possible, and if access isn't to be public use a suitable HTTP authentication scheme like HTTP Basic auth over HTTPs. Any decent web services implementation will offer authentication options or support those of the platform on which it runs. Avoid the temptation to implement your own authentication and user management at the web services layer, you will screw it up; use the auth at the HTTP layer that's already written and tested. This may require the use of something like Apache's mod_auth_pgsql, JBoss AS 7's JDBC security realms, etc. The only case I'd consider not doing proper per-user HTTP auth is where I don't need to separate my users for security reasons, I only care that it's my app accessing the server, ie if my security requirements are quite weak. In this case I'd use a fixed username/password for the whole app and possibly an X.509 client certificate if Android supports them.
Remember that no matter how you secure things, all credentials are either known to the user or can be extracted trivially from a .apk so you still have to assume anybody could access your web service methods, not just your app. Write them accordingly.
Do not just send SQL from your app over a web service call to the server and return the results as JSON. This is horrifyingly insecure, as well as ugly and clunky. Write a web service method for each individual task you want the app to be able to perform and keep the SQL in the server. Remember to use parameterised queries and be careful of other SQL injection risks. These web service methods may use one or more queries to produce a single reply - for example, you might collect a "Customer" record and all associated "Address" and "Contact" records then return the result in a nice JSON object the Android device can consume, saving numerous slow and unreliable network round trips.
No matter what you use, make sure to do your web service calls in a background worker thread and not to block the user interface. Be prepared for timeouts and errors, and for the need for retries. Test your app by simulating intermittent connection loss, high latency, and high rates of packet loss and make sure it remains usable.
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