Do you need all of them?
No, your ssh server only needs one and the client only needs to support that one type of key for ssh connections.
RSA, DSA, ECDSA, EdDSA, & Ed25519 are all used for digital signing, but only RSA can also be used for encrypting.
RSA (Rivest–Shamir–Adleman)is one of the first public-key cryptosystems and is widely used for secure data transmission. It's security relies on integer factorization, so a secure RNG (Random Number Generator) is never needed. Compared to DSA, RSA is faster for signature validation but slower for generation.
DSA (Digital Signature Algorithm) is a Federal Information Processing Standard for digital signatures. It's security relies on a discrete logarithmic problem. Compared to RSA, DSA is faster for signature generation but slower for validation. Security can be broken if bad number generators are used.
ECDSA (Elliptical curve Digital Signature Algorithm) is an Elliptic Curve implementation of DSA (Digital Signature Algorithm). Elliptic curve cryptography is able to provide the relatively the same level of security level as RSA with a smaller key. It also shares the disadvantage of DSA of being sensitive to bad RNGs.
EdDSA (Edwards-curve Digital Signature Algorithm) is a digital signature scheme using a variant of Schnorr signature based on Twisted Edwards curves. Signature creation is deterministic in EdDSA and its security is based on the intractability of certain discrete logarithm problems, so it's safer than DSA & ECDSA which requires high quality randomness for each and every signature.
Ed25519, is the EdDSA signature scheme, but using SHA-512/256 and Curve25519; it's a secure elliptical curve that offers better security than DSA, ECDSA, & EdDSA, plus has better performance (not humanly noticeable).
RSA keys are the most widely used, and so seem to be the best supported.
ECDSA, (introduced in OpenSSH v5.7), is computationally lighter than DSA, but the difference isn't noticeable unless you have a machine with very low processing power.
As of OpenSSH 7.0, SSH no longer supports DSA keys (ssh-dss) by default. A DSA key used to work everywhere, as per the SSH standard (RFC 4251 and subsequent).
Ed25519 was introduced in openSSH 6.5.