PART I: How To Log In
We'll assume you already know how to build a login+password HTML form which POSTs the values to a script on the server side for authentication. The sections below will deal with patterns for sound practical auth, and how to avoid the most common security pitfalls.
To HTTPS or not to HTTPS?
Unless the connection is already secure (that is, tunneled through HTTPS using SSL/TLS), your login form values (gasp! including your password) will be sent in cleartext, which allows anyone eavesdropping on the line between browser and web server will be able to read logins as they pass through. This type of wiretapping is done routinely by governments, but in general, we won't address 'owned' wires other than to say this: Just use HTTPS.
In essence, the only practical way to protect against wiretapping/packet sniffing during login is by using HTTPS or another certificate-based encryption scheme (for example, TLS) or a proven & tested challenge-response scheme (for example, the Diffie-Hellman-based SRP). Any other method can be easily circumvented by an eavesdropping attacker.
Of course, if you are willing to get a little bit impractical, you could also employ some form of two-factor authentication scheme (e.g. the Google Authenticator app, a physical 'cold war style' codebook, or an RSA key generator dongle). If applied correctly, this could work even with an unsecured connection, but it's hard to imagine that a dev would be willing to implement two-factor auth but not SSL.
(Do not) Roll-your-own JavaScript encryption/hashing
Given the perceived (though now avoidable) cost and technical difficulty of setting up an SSL certificate on your website, some developers are tempted to roll their own in-browser hashing or encryption schemes in order to avoid passing cleartext logins over an unsecured wire.
While this is a noble thought, it is essentially useless (and can be a security flaw) unless it is combined with one of the above - that is, either securing the line with strong encryption or using a tried-and-tested challenge-response mechanism (if you don't know what that is, just know that it is one of the most difficult to prove, most difficult to design, and most difficult to implement concepts in digital security).
While it is true that hashing the password can be effective against password disclosure, it is vulnerable to replay attacks, Man-In-The-Middle attacks / hijackings (if an attacker can inject a few bytes into your unsecured HTML page before it reaches your browser, they can simply comment out the hashing in the JavaScript), or brute-force attacks (since you are handing the attacker both username, salt and hashed password).
CAPTCHAS against humanity
CAPTCHA is meant to thwart one specific category of attack: automated dictionary/brute force trial-and-error with no human operator. There is no doubt that this is a real threat, however, there are ways of dealing with it seamlessly that don't require a CAPTCHA, specifically properly designed server-side login throttling schemes - we'll discuss those later.
Know that CAPTCHA implementations are not created alike; they often aren't human-solvable, most of them are actually ineffective against bots, all of them are ineffective against cheap third-world labor (according to OWASP, the current sweatshop rate is $12 per 500 tests), and some implementations may be technically illegal in some countries (see OWASP Authentication Cheat Sheet).
CAPTCHAs are bad for the visually impaired. The traditional copy was almost impossible to complete even for humans with perfect 20/20 vision on the first try. So don't use a traditional CAPTCHA!
If you must use a CAPTCHA, use Google's reCAPTCHA, since it is OCR-hard by definition (since it uses already OCR-misclassified book scans) and tries very hard to be user-friendly.
Personally, I tend to find CAPTCHAS annoying, and use them only as a last resort when a user has failed to log in a number of times and throttling delays are maxed out. This will happen rarely enough to be acceptable, and it strengthens the system as a whole.
Storing Passwords / Verifying logins
This may finally be common knowledge after all the highly-publicized hacks and user data leaks we've seen in recent years, but it has to be said: Do not store passwords in cleartext in your database. User databases are routinely hacked, leaked or gleaned through SQL injection, and if you are storing raw, plaintext passwords, that is instant game over for your login security.
So if you can't store the password, how do you check that the login+password combination POSTed from the login form is correct? The answer is hashing using a key derivation function. Whenever a new user is created or a password is changed, you take the password and run it through a KDF, such as Argon2, bcrypt, scrypt or PBKDF2, turning the cleartext password ("correcthorsebatterystaple") into a long, random-looking string, which is a lot safer to store in your database. To verify a login, you run the same hash function on the entered password, this time passing in the salt and compare the resulting hash string to the value stored in your database. Argon2, bcrypt and scrypt store the salt with the hash already. Check out this article on sec.stackexchange for more detailed information.
The reason a salt is used is that hashing in itself is not sufficient -- you'll want to add a so-called 'salt' to protect the hash against rainbow tables. A salt effectively prevents two passwords that exactly match from being stored as the same hash value, preventing the whole database being scanned in one run if an attacker is executing a password guessing attack.
A cryptographic hash should not be used for password storage because user-selected passwords are not strong enough (i.e. do not usually contain enough entropy) and a password guessing attack could be completed in a relatively short time by an attacker with access to the hashes. This is why KDFs are used - these effectively "stretch the key", which means that every password guess an attacker makes causes multiple repetitions of the hash algorithm, for example 10,000 times, which causes the attacker to guess the password 10,000 times slower.
Session data - "You are logged in as Spiderman69"
Once the server has verified the login and password against your user database and found a match, the system needs a way to remember that the browser has been authenticated. This fact should only ever be stored server side in the session data.
If you are unfamiliar with session data, here's how it works: A single randomly-generated string is stored in an expiring cookie and used to reference a collection of data - the session data - which is stored on the server. If you are using an MVC framework, this is undoubtedly handled already.
If at all possible, make sure the session cookie has the secure and HTTP Only flags set when sent to the browser. The HttpOnly flag provides some protection against the cookie being read through XSS attack. The secure flag ensures that the cookie is only sent back via HTTPS, and therefore protects against network sniffing attacks. The value of the cookie should not be predictable. Where a cookie referencing a non-existent session is presented, its value should be replaced immediately to prevent session fixation.
Session state can also be maintained on the client side. This is achieved by using techniques like JWT (JSON Web Token).
PART II: How To Remain Logged In - The Infamous "Remember Me" Checkbox
Persistent Login Cookies ("remember me" functionality) are a danger zone; on the one hand, they are entirely as safe as conventional logins when users understand how to handle them; and on the other hand, they are an enormous security risk in the hands of careless users, who may use them on public computers and forget to log out, and who may not know what browser cookies are or how to delete them.
Personally, I like persistent logins for the websites I visit on a regular basis, but I know how to handle them safely. If you are positive that your users know the same, you can use persistent logins with a clean conscience. If not - well, then you may subscribe to the philosophy that users who are careless with their login credentials brought it upon themselves if they get hacked. It's not like we go to our user's houses and tear off all those facepalm-inducing Post-It notes with passwords they have lined up on the edge of their monitors, either.
Of course, some systems can't afford to have any accounts hacked; for such systems, there is no way you can justify having persistent logins.
If you DO decide to implement persistent login cookies, this is how you do it:
First, take some time to read Paragon Initiative's article on the subject. You'll need to get a bunch of elements right, and the article does a great job of explaining each.
And just to reiterate one of the most common pitfalls, DO NOT STORE THE PERSISTENT LOGIN COOKIE (TOKEN) IN YOUR DATABASE, ONLY A HASH OF IT! The login token is Password Equivalent, so if an attacker got their hands on your database, they could use the tokens to log in to any account, just as if they were cleartext login-password combinations. Therefore, use hashing (according to https://security.stackexchange.com/a/63438/5002 a weak hash will do just fine for this purpose) when storing persistent login tokens.
PART III: Using Secret Questions
Don't implement 'secret questions'. The 'secret questions' feature is a security anti-pattern. Read the paper from link number 4 from the MUST-READ list. You can ask Sarah Palin about that one, after her Yahoo! email account got hacked during a previous presidential campaign because the answer to her security question was... "Wasilla High School"!
Even with user-specified questions, it is highly likely that most users will choose either:
