What do you like best?
I love that I can create forms for people to sign up (like to volunteer for an organization), can have both free and paid events, can do mass email without extra costs, and can even do memberships (free and paid). If you have an organization where you have employees/volunteers contacting people, you can even keep track of all of that on their record; This allows the next person to know who spoke with them last, what was discussed, if you missed them and left a message, etc.
What do you dislike?
Sometimes upgrading can be difficult and you end up having to start with a fresh database and import in all your SQL. I also run into issues sometimes and find it difficult to get support for it.
Recommendations to others considering the product
I highly recommend this product. It is free to use, although you might need a web designer to help you get it set up and ready to be used. I use it with Drupal, but you can also use it with other CMS systems like WordPress. Once set up, your only costs would be the web server where it is hosted and when you occasionally need a webmaster to make changes, do updates, etc. This could mean savings of hundreds - or thousands - a year when compared to paid services.
What business problems are you solving with the product? What benefits have you realized?
We use it for a number of tasks:
For non-profits and political groups, we collect volunteer information, take online donations, have sign-ups for lawn signs and to endorse the candidate, have sign-ups for free and paid events, and send out bulk email. The ability to do events is great, as you can set up your fundraisers on there and then take money via Paypal, Authorize.net, etc. Many non-profits do their membership programs through CiviCRM as well.
For businesses, it's often times used for paid events, memberships, and bulk email.