Introduction

Coveo is a provider of intelligent and predictive search technologies with integrated plug-ins for Salesforce, Sitecore, Google Apps for Work, Atlassian, JIRA, Office 365, YouTube, Lithium, Jive, Dropbox, and more APIs also allow for custom integration with other applications.

Coveo was founded in 2005 with 30 employees in Quebec City and in Palo Alto, California. Coveo's intelligent search technology adds the value of rich content and insights to CRM, customer service applications, intranets and websites. Coveo securely connects with your legacy and cloud systems, and provides unified search, dynamic 360-degree views of information, and contextual, proactive recommendations of relevant content and experts using powerful analytics.
Wikipedia

Improve Sitecore Search Experience

Search is kind of core for any site now a days. People are smart rather drilling through the site they want someone to lookup for the information that they are looking for. It really shines when you have a huge amount of data. Typical search would create index per data source. Coveo makes your life easy that can combine multiple data source into single search index.
Coveo has the following advantages. I will be focusing on the cloud option for Sitecore but most of it is supported in on-premise too.

Integration

Plug and play support for most of the platforms. Cloud versions in hosted in industry best cloud provider AWS.

Unified index

Major benefit since each platform has its own search. Sitecore depends on Lucene or SOLR. SharePoint works on internal search(FAST). A Single index based on Coveo helps to search across different data sources.

Powerful JavaScript UI

Coveo owned UI framework is light and fast. Supports clean html rendering using data attributes. Templating frameworks are supported like handlebar or Underscore tmeplates.
The Coveo JavaScript Search Framework allows developers to easily deploy and customize a feature-rich client-side search interface in any web page, application, or standalone site to present results from a Coveo Cloud or on-premises index. 

WYSIWYG UI

Search pages can be designed on the fly in a browser with drag and drop support. No coding needed design your own UI right from the browser.(Cloud UI)

Security Aware

Coveo shines when it comes to security. It has a security cache that has all the security information from all sources. When a user queries it can chain the security and return the results according to the user permission. The trimming is done before the search results are returned. This can be disabled too for public facing sites considering the peformance.

Machine learning

Coveo machine-learning capability, analytics-based machine learning service analyzes your users’ search activity as well as your diverse content to always surface the best, most relevant answers and information for everyone. We can fine tune the learning model in a scheduled frequency. Understands the user intent and provide relevant query suggestions. The Query pipeline can be used to fine tune the results with boosting the most relevant ones. Type ahead support, Related results and context based recommendations.

Analytics

Essentially, events performed by end-users in Coveo search interfaces (and optionally in other web page types) are recorded by the usage analytics service in the Coveo Organization. We can then use the administration console Analytics pages to learn what end-users do by reviewing usage events, metrics, and dimensions in explorers and dashboards. This helps in analyzing the user search and improve the existing content or missing content. The best part it can be talk to the XDB the core of Sitecore analytics to drive the search results.

Sitecore Experience Database (xDB) Integration

Leveraging the comprehensive profile and activity data managed within Sitecore’s xDB, Coveo for Sitecore makes it possible for marketers to easily tailor personalized search experiences and content pages for each unique website visitor. By utilizing the insights the Sitecore xDB captures about each site visitor - including select demographic information, prior activity on the site, and past exposures to email and other marketing initiatives - marketers can use Coveo for Sitecore to offer each site visitor the products and content most relevant to them at that precise moment in their customer journey. And because it is fully integrated into the Sitecore platform, marketers can build and manage these experiences with the Sitecore tools they use to manage their entire website.

Versions

Coveo provides a local install for free with limited queries and capabilities. Cloud comes in multiple versions. It also offers one month cloud trial version with no credit card requirements.

Ref: Coveo current pricing information

Support

The support is very responsive and Developer friendly, they are active on slack channel that gives you opportunity to talk to the evangelist and Coveo inhouse developers. They have One of the best documentation I have used in recent times. The community forum is great information source.

Competitors

I came across the following competitors providing search solutions. Officilaly Sitecore backs Coveo integration.
Swifttype
Azure Search