They're the same thing. Alexa is the "intelligence" and voice for the Echo. Echo has a few different types - the Echo, the Tap, and the Dot. Echo is voice activated, Tap has to be, well, tapped to activate, and the Dot is about the size of a hockey puck and is voice activated. I own both an Echo and a Dot.
The only alternative I know if is the Google Home but I don't know much about it.
It's an internet enabled speaker, basically. Echo is the big one. Dot is the little one.
I have the Echo Dot. It's pretty cheap, has an okay little speaker, but I don't use it as a a speaker. I use the home stereo. It gets the most usage in our house as a client for Spotify, setting timers and alarms, and using the TV when I can't find the remote. I use a Harmony hub with it and it works really well... except for a couple of small flaws. It's basically Siri for your house. Google Home is basically the same thing. Google is a bit more expensive... it really excels when interfacing with Google Search and ChromeCast. (Surprise, Surprise) Alexa has basically no google interface because Google and Amazon pretend the other doesn't exists... which is a annoying. It takes a bit of work to get either set up... and I'm slowly replacing switches in my house so I can get the lights hooked into Alexa... If you want to do more than listen to music on the internet and play Jeopardy... it requires a bit of work. The voice recognition is very very good though.
Echo is the ecosystem, Alexa is actually the wake word - so you say 'Alexa' to wake your Echo and have it listen to a question or command. You can set the wake word to be 'Echo' or 'Computer' (think Scotty) if you want.
Depending on how much you want to spend and how much work you want to do, the Echo can be integrated into just about everything. I have it so I can turn on and off lights and lock doors along with playing music and answering questions. The dot is small, inexpensive and isn't intended to be a music player although it does have a speaker adequate for 'talking' responses. The Tap is portable and originally as posted above, you had to tap it to make it listen - they have since made an update and you can set it to listen on the wake word without tapping it. It's pretty great to be able to bring it outside for music or to tell it to turn on lights in the backyard, etc. It lasts about 8 hours on charge (or forever left on the charging base). The full Echo is the same as the others, just a better speaker capable of reasonable music sound. They have also released the Echo Show which has a screen. The biggest use case I've found is you can link it to the Ring doorbell so you can see who is at your front door. It also displays song lyrics which is nice, but not worth the price upgrade from the full Echo.
I'm a big fan of the system but it does take work and $$$ to make it really useful. Zwave door locks, lighting controllers/switches/bulbs, Nest or Ecobee thermostats and Ring doorbells and cameras all add up quickly. One thing I really like about it as it has consolidated various isolated home automation implementations I had done. It does compete with the Google product. The main thing here is Echo had about a year + headstart so many skills have been written for it.Basically, Amazon released it as a test kind of device and then found out later, along with Google and Apple, how much people wanted it. For about a half year, you had to be invited to get one. It really appears to me it was a total fluke product that nobody knew they wanted/needed.