Amplitude is an event-based analytics tool which gives you comprehensive analytics across your web and app users. You can add events like "Signed up" or "Clicked on Login Button" and then can create a graph on the event to know how many people are actually doing that event. They also have a realtime user dashboard where you can see what live users are doing but as it is in tabular format, it is not easy to track someone from that. You can also add events to your emails to know the open rate or click rate.
They do not have a published pricing plan and their pricing page says to contact them to know about pricing. Although, it is mentioned that they have a free plan which will track up to 10 million actions per month. Beyond that it becomes paid and it depends upon the number of actions you wish to keep record of. Some resources mention that you have to pay almost $2k per month beyond that.
Amplitude's headoffice is located in San Francisco and has offices in New York, London, Paris, Amsterdam, Seattle, Austin, and Singapore.
Yes, Amplitude works with your mobile platforms like iOS and Android Apps. You can just add their SDK and keep on adding events wherever you please. Once the events are added, it will start collecting your data for analysis. You can also send identify calls in order to unify the data across web and mobile platforms where th data of any user who is logged in will get unified.