I am looking for an Android application that can record the path, as well as the speed at each point of the path when I drive.
For example, the Android app RunKeeper can do it for jogging:
Any price is good.
As commenters have stated, there are any number of apps that could do what you require to a greater or lesser degree, including Runkeeper itself.
After trying several, the one I have settled on is Cycledroid.
As the name suggests, it is aimed at cyclists. However, it exports the information that you require in GPX or KML format.
It doesn't try to match your performance with others (like Strava), so you don't upset cyclists by beating their segment records by 50mph. It has a simple interface, uncluttered by calorie counts and similar, which aren't relevant to driving.
The Trip Recording plugin allows you to record your trip and save it as a GPX file. It uses your phone's GPS to track your movement. You can view the route you've travelled, send it to OpenStreetMap project or upload it to external services.
That would be an all-in-one solution since you could use just one app for your (offline) map viewing, navigation, POIs and trip recording, all based on OpenStreetMap and Wikipedia data. It is not specific for driving, it can be used for activities such as running, walking, cycling, etc..
Some of the data recorded:
The app is open source, actively developed and it's available in two versions: OsmAnd (free) and OsmAnd+ (paid). With the paid version you support the project, fund the development of new features, and receive the latest updates (you can get an unlimited number of map updates). The Trip Recording plugin works fine with both the free and paid versions.
Here are three basic screenshots from the website: