You can find all about Let's Encrypt initiative at their website. The most current instruction can be found at EFF's certbot site.

This one is for Ubuntu 14.04 (trusty) and pursues the easier-to-use option. I'm assuming that you have a sudo access, although that isn't an absolute requirement.

Installation

Last tested on Ubuntu 14.04.2 LTS (2016-06-27)

Note: On Ubuntu 16.04.01 LTS (xenial), you can just run apt install letsencrypt and use letsencrypt in place of certbot-auto command in the following instruction. It's the equivalent program, so there doesn't seem to be any issues with parameters used here.

Download the executable and make it executable.

$ sudo wget https://dl.eff.org/certbot-auto
--2016-06-27 18:36:18--  https://dl.eff.org/certbot-auto
Resolving dl.eff.org (dl.eff.org)... 173.239.79.196
Connecting to dl.eff.org (dl.eff.org)|173.239.79.196|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 44115 (43K) [text/plain]
Saving to: ‘certbot-auto’

100%[======================================================================================================================================================>] 44,115      --.-K/s   in 0.001s

2016-06-27 18:36:18 (67.8 MB/s) - ‘certbot-auto’ saved [44115/44115]

$ sudo chmod a+x certbot-auto

I prefer to have this type of executable in /usr/local/bin/ folder. It will make it available for other users as well as make it easy add as a cronjob. The ownership is already correct if you used sudo.

Typing sudo certbot-auto --apache in CLI will get you to an interactive menu that will list out all of your domains on Apache2 and will easily generate certificates and even add those Apache directives in the respective virtual domain configuration files (not 100%, but works most of the time).

certbot-auto creates a folder in /etc/letsencrypt/ as a default.

List the certificates

$ certbot-auto certificates

Adding more domains

Last tested on Ubuntu 14.04.2 LTS (2016-06-27)

After the initial installation, if you need to add more domains you can do it directly from the CLI.

$ sudo certbot-auto run --apache -d mydomain.net

Configuring to auto-renew certificate

Last tested on Ubuntu 14.04.2 LTS (2016-06-27)

certbot-auto can also auto-renew certificates by adding a command as a cronjob.

0 1,13 * * * /usr/local/bin/certbot-auto renew --quiet --no-self-upgrade

Use crontab to update the cron jobs, and add the above line.

$ sudo crontab -e

Adding multiple domain names for a certificate

Last tested on Ubuntu 14.04.2 LTS (2016-06-27)

You can use one certificate for multiple domains. The certificate is generated, but you need to install it yourself.

$ certbot-auto certonly --webroot -w /srv/www/mysite.com/ -d www.mysite.com -d mysite.com -w /srv/www/blog.mysite.com/ -d blog.mysite.com

IMPORTANT NOTES:
 - Congratulations! Your certificate and chain have been saved at
   /etc/letsencrypt/live/mysite.com/fullchain.pem. Your cert
   will expire on 2016-09-26. To obtain a new or tweaked version of
   this certificate in the future, simply run certbot-auto again. To
   non-interactively renew *all* of your certificates, run
   "certbot-auto renew"
 - If you like Certbot, please consider supporting our work by:

   Donating to ISRG / Let's Encrypt:   https://letsencrypt.org/donate
   Donating to EFF:                    https://eff.org/donate-le

Getting certificates for a separate server using manual mode

Last tested on Ubuntu 14.04.2 LTS (2016-06-27)

This will start an interactively, manual mode.

$ certbot-auto certonly --manual -d test.com -d www.test.com

Revoking and/or deleting certificates

Last tested on Ubuntu 14.04.2 LTS (2016-06-27)

The optional reason flag can have the following values: unspecified (default), keycompromise, affiliationchanged, superseded, and cessationofoperation.

$ certbot-auto revoke --cert-path /etc/letsencrypt/live/bamboo.domain.com/cert.pem --reason cessationofoperation

Deleting certificate files

$ certbot-auto delete --cert-name domain.com

Generating a certificate for wildcard subdomains

Last tested on Ubuntu 16.04.5 LTS (xenial) (2018-09-09)

You need to be able to update the DNS zone file for the domain. Usually, this is referred to as DNS records, and many registrars have a way for you to update the records. For this, you should be able to add a TXT record.

$ sudo certbot-auto certonly --manual -d *.mydomain.com --agree-tos --no-bootstrap --manual-public-ip-logging-ok \
    --preferred-challenges dns-01 --server https://acme-v02.api.letsencrypt.org/directory
Saving debug log to /var/log/letsencrypt/letsencrypt.log
Plugins selected: Authenticator manual, Installer None
Obtaining a new certificate
Performing the following challenges:
dns-01 challenge for mydomain.com

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Please deploy a DNS TXT record under the name
_acme-challenge.mydomain.com with the following value:

w_DD-eTEIk4VxoNFBm6VbLTkNkcD1pml_DYpzIRtK7M

Before continuing, verify the record is deployed.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Press Enter to Continue
Waiting for verification...
Cleaning up challenges

IMPORTANT NOTES:
 - Congratulations! Your certificate and chain have been saved at:
   /etc/letsencrypt/live/mydomain.com/fullchain.pem
   Your key file has been saved at:
   /etc/letsencrypt/live/mydomain.com/privkey.pem
   Your cert will expire on 2018-12-08. To obtain a new or tweaked
   version of this certificate in the future, simply run certbot-auto
   again. To non-interactively renew *all* of your certificates, run
   "certbot-auto renew"
 - If you like Certbot, please consider supporting our work by:

   Donating to ISRG / Let's Encrypt:   https://letsencrypt.org/donate
   Donating to EFF:                    https://eff.org/donate-le

Resources

Complete guide to configure SSL on Nginx with Let’s Encrypt for Ubuntu/CentOS/RHEL (Snapshot)