If you are not yet using Maven or Gradle, you can include osmdroid by downloading directly from Maven Central.
You have a choice between using an AAR (for >=5.0) or JAR (for < 4.3) in your libs folder.
For versions < 4.3, you also need slf4j-api and slf4j-android.
Version >= v5.0
If you're still using Eclipse without a backing building system (maven/gradle), you'll probably want to google around on how to use AAR files with Eclipse. By now, most people have probably moved on to Android Studio with gradle, since it's really the only supported solution. Some users have reported success by unzipping the AAR and the classes jar files and then including them in the classpath.
Good luck!