These are my thoughts about (so far) three third-party apps that work with Zotero to view, edit, and read from your Zotero Library with mobile devices. ZotPad and PaperShip are pretty much direct competitors with each other; ZotFile is a little different.
Devices: iPad, iPhone, Mac (I only looked at the iPad version)
Platforms: works with Zotero and Mendeley (at this point I've only looked at the Zotero version) Supports access to saved PDFs on Zotero.org, Box, and other webDAV servers. PDFs can be set to always download (not recommended -a big memory hog on your device!) or download on demand via the gear (settings) icon.
Cost: Now free, with an optional $4.99 in-app purchase for annotation tools
Viewing and Searching:
Annotation and Searchability
PaperShip includes a highlighter (with changeable colors) and you can also purchase in-app additional annotation tools: writing on the document, pop-up notes (you can change the designated icon using colors and shapes such as stars or checkmarks ), drawing tools , insert voice, image insert, stamps,. and grouping. The PDF tools follow PDF standardization, so searchable annotations can be searched outside of PaperShip (keeping in mind the usual PDF thing that functionality depends on how the PDF was annotated and what you ae viewing it in; generally: 1. only selectable annotations can be searched, and 2. searching may be limited if you are searching in-browser versus on a separate PDF-reading application) The tools all have nice formatting options via the Inspector option. (It's really worth the $5).
Searching within the Collections Pane: titles, authors, and tags
Searching within PDFs: PDF text and searchable annotations (see above)
Screenshots cannot be annotated and cannot be searched (NOTE: screenshots aren't actually screenshots - they are live windows from the website which was captured, which may actually limit their utility).
Sorting and Navigation
Items in your library and collections can be sorted by: title, author, journal, publication date, modification date, and "Altmetric Score" (see below).
Items with viewable attachments (PDF or screenshot) can be moved or copied to Collections via the folder icon (this means that items with only bibliographic metadata cannot be copied or moved).
Syncs with both Zotero.org and Mendeley. Pull down on the top of the Collections pane to sync. It can be slowish...
Sharing and "Open in..."
Using the usual iPad icon you can share citations via Twitter, email, and Message. Items can also be opened in other PDF reading and annotating applications such as PDFReader or IAnnotate - if any changes are made to them, your PDF can be exported back to PaperShip with the annotations intact, and those annotations can be edited or deleted via PaperShip's own annotation tools.
Help is available via the in-app gear icon (email support, FAQ, links to PaperShip website, blog, and Facebook pages).
Lorena's thoughts (as of 8/11/2014)
This is a pretty spiffy app. It's really worth the $5 to purchase the in-app annotation tools. I like that it works with both Zotero and Mendeley, and I'm interested to see how the partnership with altmetric.com (NOT altmetrics.org) will pan out. ITunes reviews are mostly positive, with some complaints about Mendeley syncing and general syncing time.
working on it, but as of 9/2/2015 it does not seem to have been updated since early 2014, and App Store reviews are not very good.
10/15 Update: I recommend PaperShip (see box to the left)
Not precisely a mobile app, but it provides some cool functionality that can help you use your tablet for opening a PDF in your Zotero library, annotating it using your PDF apps (I.e. iAnnotate or PDFExpert). and then sending it back to your Zotero library with annotations. (that ZotFile can then extract into a Zotero Note). See the "Using ZotFile for Tablet Syncing and More" tab on this guide for more on this.