Full Disk Encryption also referred to as FDE encrypts an entire storage medium, most notably Hard Disk Drives (HDDs) which is where FDE gets its name. Several forms of FDE exist today including but not limited to, BitLocker, LUKS/dm-crypt, and VeraCrypt. When a storage medium is encrypted using FDE the all the data including file metadata is encrypted, a properly encrypted drive discloses no information to an adversary other than the fact that it may be encrypted. The security of Full Disk Encryption when done correctly is decently high. Due to the nature of FDE encrypting everything, it is difficult to even infer or guess the contents of the drive without decrypting the drive. When used with a strong memorized secret such as a random Diceware passphrase, it is nearly impossible to decrypt a drive without getting the user to disclose the memorized secret or though malware such as a keylogger. Security can also be added by using a key file, without the key file the drive cannot be decrypted. Preferably, an encrypted drive should be overwritten with random data before encryption, and memorized secrets should be of sufficient length.
File encryption also referred to as File Based Encryption (FBE) is the process of individually encrypting files. There are several forms of file encryption, some with their own drawbacks and advantages. File encryption can be applied to individual files or chunks of files two notable examples of this are Picocrypt and CryFS, both work differently but both are forms of File Encryption. File encryption like Full Disk Encryption is very hard to get around when it is done correctly. However, there are some ways to infer what the encrypted file may contain. File encryption often times does not encrypt the metadata of files such as their names, sizes, and directory structures. This metadata may allow an adversary to infer what the encrypted file may be. Some file encryption schemes will account for this, such as CryFS which will hide most of the metadata.
File encryption does have its uses in areas where Full Disk Encryption is not suitable for use, such as in the cloud. File encryption also commonly includes authenticated encryption, which can detect malicious modifications to encrypted data. This is usefully for when any file is transmitted through the internet and may be intercepted by an adversary. It not only maintains confidentiality, it maintains integrity of the data. Unauthenticated encryption especially when using cipher modes such as CBC can lead to attacks such as the Padding Oracle Attack, authentication using something like HMAC or AEAD cipher prevents this attack from occurring.
Message encryption such as the Signal protocol and OMEMO are designed to keep your messages secure so only you and the intended recipient can read them. These protocols make use of a variety of cryptography, including symmetric ciphers such as AES and asymmetric public key cryptography such as Elliptic Curve Cryptography(ECC) and Curve25519. As long as the keys have not been swapped out during the key exchange (can be verified by comparing fingerprints (OMEMO) or a safety number (Signal protocol)) and neither of the endpoints are compromised. The messages are almost guaranteed to not be read by anyone else but you and your recipient. Security can further be increased by deleting messages after they have been read.
Choosing the right cipher is not as important as you think. Most of the time in properly made software a secure cipher is used by default such as AES or ChaCha20. However, you may want to be aware of different ciphers available to you and which ones you should definitely avoid. Most modern ciphers are not currently broken and should be OK to use. This includes the AES finalists such as Serpent, Twofish, RC6, MARS, and the AES winner Rijndael. The cipher of choice is AES (slightly modified version of Rijndael) because it has had the most analysis and testing done on it and proper implementations have practically never been broken and probably won't be broken for a very long time. AES also has hardware acceleration making it extremely fast on most processors, causing little to no performance impact on the device it is used on. ChaCha20 has now seen more use especially in TLS, it is also fast making it a great alternative to AES in some systems. Serpent, Twofish, RC6, and MARS are also good choices but shouldn't be your first choice. Like AES they are not broken but have received way less analysis and do not have any hardware acceleration so they are often times slow, however if AES is broken, these ciphers are a safe choice.
Here is a list of ciphers that should be avoided because they have a severe weakness or have been broken.
*Kuznyechik's hidden S-Box structure has caused suspicions of a backdoor, however this was the same case for DES. The NSA had known about differential-cryptanalysis and modified the DES S-Boxes to be resistant to it, without publicly disclosing the reason. The creators of Kuznyechik could know about some new cryptanalysis technique and could have done the same to their cipher, this is more likely as the S-Box structure doesn't seem to actually weaken the cipher.