Mobile Options for your Salesforce Community
1) Responsive Web App — Our Lightning community is designed to be mobile responsive and will render well on any device form factor like mobile, tablet etc.
This path is the least work since all our standard components behave like this out of the box.
2) Native App — At the other end of the spectrum you can build a fully branded native application on iOS or android, use our mobile SDK to provide auth, offline and access to full Salesforce back end as needed.

Footnote — some Salesforce partners and SIs have templated apps / accelerators in this areas also.
Find out more about the Salesforce mobile SDK : https://developer.salesforce.com/developer-centers/mobile/
3) Hybrid App- Architecturally this is a combination of approach (1) and (2) with main benefit of reducing implementation time while still getting most of the benefits of a native app. Salesforce are planning to have a hybrid app option available for communities in the next 12 months. Conceptually it will be similar to MySalesforce and will act as Hybrid app container for your Salesforce Community
4) Salesforce Mobile — It is possible for external community users to use their credentials with the standard Salesforce Mobile App. In my view not a great option for external users since its harder for you to control the UX and branding plus there are some extra setup steps for users.
Essentially when you access Salesforce as an internal user you always start on login.salesforce.com however this step is different for community users who typically have a dedicated login URL like portal.mycompany.com
Since the app doesn’t know about this app URL in advance the user needs to configure it in the app.
On the login screen click the gear icon:

and then select the option to add a new connection. Here you need to specify the login URL for the community.

Below are the help and training details on using Salesforce Mobile with community users.
How to setup your community user with the Salesforce Mobile App: https://help.salesforce.com/articleView?id=networks_access_in_salesforce1.htm&type=5
What’s different between the internal Salesforce App experience and the Community App experience:
Recommendation
My recommended approach would be generally to start with option 1 — a mobile responsive website / portal — easiest to implement and affords you to flexibility to experiment with use cases and experiences. When Salesforce releases the hybrid app option (3) you will be able to easily take advantage of the enhanced capabilities there if needed.
If you already have a mobile app and just need to plug some salesforce capabilities into that app then the mobile SDK makes a lot of sense for this use case. You may also want to investigate Salesforce mobile snap-ins for this scenario : https://developer.salesforce.com/page/SnapinsMobile