These are the tools that I use every day as a product owner (and a few that I used at the last company, but found to be quite useful). If you have a question about any of these tools, feel free to reach out to me on twitter or shoot me an email.
Project Management

Wrike
A task management tool for teams.

JIRA
One of the most popular software programs for developers to manage user stories.
Team Communication
Real-time direct messaging and chat rooms. These are both starting to offer new features like screenshare and audio chat. I recommend using at least one of these two systems:

Slack
Possibly the best communication tool for teams that there is.

Hipchat
Same as Slack. It is hard to say which is better, they’re neck-and-neck. Either option is better than no option (and WAY better than relying on Google Hangouts or something like that).
Audio + Video Chat
The important thing to remember (especially with remote workers) is that one of these is always going to be crappy on the day of your meeting. Always have at least two lined up so that if one is flaking out, your team can quickly switch without wasting precious time!

Google Hangouts
I HATE using for chat, but an important member of my battery of voice chat software.

Skype
The old stand-by. I also have a Skype phone number that has been quite useful since I’m remote (and people think I live in Austin!)
Online Visual Collaboration
(because screen-share isn’t interactive, and people will fall asleep).

Mural
Online pin-up boards. I love using these as a visual workspace when I’m on calls with teammates.
Personal Scheduling

Google Calendar
Better than Outlook – sorry.

Calendly
Create links so people can schedule meeting with you in pre-determined meeting slots.
Shared Assets and Information
A big part of being effective is knowing what to store where. For anything that can possibly be saved in a wiki (meaning, not files) I use:

Confluence
Confluence is a pretty robust wiki – I use it for everything. Not the easiest to set up and maintain, but its getting better, and SUPER robust once you learn it.
For files, I personally use two cloud services, and our business uses a Dropbox business account:

Google Drive
Google Drive is nice for it’s integration with Google Docs. It is more robust (read: confusing) than Dropbox.

Dropbox
More cloud storage. I find Dropbox less confusing for sharing, but confusing in other ways.
Digital Prototyping and QA

Invision
A fantastic (and fast) online software for creating and sharing prototypes of software.

Hockey App
An app for hosting and sharing builds of phone apps. We used this at my last company, and it was pretty useful.

Telecine
A simple, solid android app for recording your phone screen. I use this when I’m doing QA for stuff on mobile.