The content, and to some extent the design, of an application is protected by copyright. If anyone copies the content, or closely imitates the design, the original creator (or whoever has bought the rights, if the creator sold them) can sue for copyright infringement. Copyright is international, and applies almost everywhere in the world. All those countries belonging to the Berne Copyright Convention grant to citizens of other countries that belong the same rights as to their own citizens. Almost every country in the world is now a member.
Some additional protections can be obtained in the US if the creator chooses to register the copyright. This provides statutory damages when actual damages are hard to prove, and has other benefits. But there is a fee, and this is not required for copyright protection.
Copyright does not protect the name of an application. Instead, that can be done via Trademark protection. Trademarks are specific to individual countries. They can be established by simply using the name in commerce, that is by selling the product or advertising it for sale. However, one often obtains stronger protection by registering the trademark. Each country has its own procedures. For the US rules visit the USPTO site. Logos can also be protected as trademarks. A fee is charged for registration.
Note that trademark protection is limited to the field or industry in which bit is used. For example, a trademark on a game called "Great Times" would not prevent the use of that as the name for a resort. Unique invented names are generally stronger trademarks than are descriptive names. Preparing a proper trademark application can be tricky, and many people hire an expert to assist. But basic trademark protection can be obtained simply by putting a product or service on the market.
Neither copyright nor trademark protection will protect the basic idea of a game or other product. But when a detailed, point-by-point copy is made, there may be infringement.
The owner of a copyright or trademark must do whatever enforcement s/he choses. The police will not do it for the owner. The owner must find out who is infringing the copyright or trademark, and take (or threaten) legal action. The owner may hire people to help with this. Actually filing a court case will usually involve a lawyer. Procedures will be different in different countries.
Both trademarks and copyright protection apply to applications on different platforms, or sold in different stores. Making a copy (or near copy) of an android game for the iPhone and selling it on a different eStore will be a copyright infringement, and if the name is the same or similar enough to cause confusion, this will be a trademark infringement.