How to use GPS/Maps without an Internet connection? I need a way (possibly a third-party application) that allows me to use maps and GPS without an Internet connection.

I don't use it by car, just for walking in cities. It would be nice if I could just copy all data relative to a particular city before going there for tourism.

The Google Maps app does not store maps reliably, so it is risky to rely on it without an Internet connection.
Note: many of these applications use map data from a really cool free map project called OSM (OpenStreetMap), which anyone can contribute to. OSM is constantly improving, but in many areas it may lack information like house/building numbers.

Free:


Google Maps


Google Maps does have some offline functionality (including offline navigation as of version 9.17), but the size of the area you can download is limited (you can't download an entire US state, for instance) and some countries or areas aren't available for download. See here for instructions: Download a map and use it offline

HERE Maps


Unless you're looking to support OSM, this is probably the best free option out there. It works offline, has maps in nearly 200 countries, and turn-by-turn navigation in over 100 countries. Features include full street addresses, points of interest, alternate routes, traffic, public transport, and speed limits. Notable features that are missing: ability to add multiple stops to a route, lane assist, rerouting based on traffic, and ability to navigate to contacts' addresses.

OsmAnd


Open-source, uses data from OSM.  Currently only has 2D navigation, but one of the better free options available.

MapFactor


OSM-based offline navigation.  Fairly basic but promising. TomTom maps available as an in-app purchase.

Navmii (formerly NavFree): North/Latin America, elsewhere


Ad-supported. Allows you to download maps for offline navigation, also OSM-based.  OSM has maps for the whole world, but for some reason Navmii only has maps available for certain countries, mostly in Europe and North America.

Many others based on OSM here: Android - OpenStreetMap Wiki


Paid:


CoPilot Live Premium: USA, elsewhere


CoPilot Live is a very nice, full-featured app, which will let you download maps ahead of time and use GPS offline. I ended up using it because the US version was so cheap compared to the other paid navigation apps, but I've been very pleased with the quality, features, customer support, and free app and map upgrades. It has all the features I could want in a GPS app, the only downsides being you have to have a paid subscription for traffic data (the paid app includes a free one-year subscription), and you will have to buy new maps if you travel outside your purchased area.

Other paid apps that I haven't used:


Garmin viago
Magellan SmartGPS
NDrive
Scout (formerly Skobbler): USA, elsewhere
Sygic
TomTom



There are detailed reviews of many Android GPS apps here: Android Sat Nav Apps