The terms are from the portion of the tax code, e.g. section 401(k).
401(k)'s are typically done by private (i.e. non-governmental) companies. 401(a)'s are typically done by governmental, educational and non-profits.
401(a)'s have more control than 401(k)'s in terms of who is eligible, mandatory contributions, etc. The IRS also makes it MANDATORY that institutions contribute to their employee(s) 401(a) plans. This is one of the reasons that many educational and non-profits do not offer 401(a) plans, but instead go the 403(b) or 457(b) route.
403(b) outlines plans associated with public schools and non-profit organizations (e.g. 501(c)(3) organizations. 403(b) plans do not require matching by the institution. The IRS has further restrictions on the investments, e.g. annuities, mutual funds. In the old days, 403(b)'s were typically annuities (yuck), but in recent years many funds (including my employers) are more flexible.
457(b) plans are similar to 403(b) plans but there are differences. The 457(b) plan rules state that you don’t have to pay a 10% tax penalty if you resign or retire before age 59.5 and need to withdraw money from your account. However, the 403(b) plan charges the 10% early withdrawal penalty.
Most importantly, you can contribute to BOTH a 457(b) and 403(b). That is, the limits are separate. Let's say you are 55, you can (in 2019) contribute $25,000 ($19,000 + $6,000 catch up) to your 457(b) AND contribute $25,000 ($19,000 + $6,000 catch up) to the 403(b). These rules were changed about 15 or so years ago to allow this:
https://www.irs.gov/pub/irs-drop/rp-04-56.pdf