- Chain Algebraic
Description:Mostly used in old single line display calculators, some business calculators, desktop calculators. Does not follow operator precedence. sin90 is entered in RPN style:  [SIN].
Some Calculators that uses it: Casio fx-570, HP10bll
- Equation operating system (EOS) style algebraic (Single line)
Description: Mostly used in calculators (scientifics) before dual line display calculator became common. Follows operator precedence and you enter sin90 as [SIN]  [=]. Input the way as that the equation is written. May or may not show the equation you type, depending on manufacturer’s implementation.
Marketing terms: Casio’s V.P.A.M (Visually Perfect Algebraic Method), Sharp’s D.A.L (Direct Algebraic Logic).
Some Calulators that uses it: Casio fx-992s, Sharp EL-503W
- EOS style algebraic (Dual line), also known as linear mode
Description: Most modern calculators (scientifics & even some business calculators) uses dual line display. Follows operator precedence and you enter sin90 as [SIN]  [=]. Input the way as that the equation is written. Equation that you evaluated is shown in the top line, while the answer is shown on the bottom. Some of them have a playback feature that let you recall your previous calculation. A few calculators have dual mode that can switch between linear mode and textbook mode (see below).
Marketing terms: Casio’s S.V.P.A.M (Super Visually Perfect Algebraic Method), ?Sharp’s Advance D.A.L (Direct Algebraic Logic)?.
Some Calculators that uses it: Casio fx-115MS, HP 35s, Sharp EL-506W, TI-30X llS
- EOS style algebraic (Textbook mode)
Description: Some of the newer scientific calculators are equipped with this textbook mode where the display of equations resembles equations in textbook, i.e where you input an equation as: instead of πr^2. All uses full dot matrix displays and Casio’s models can revert back to linear mode. Not too sure about Sharp and TI though. Input style is very similar to typing equations on Microsoft Words.
Marketing terms: Casio’s NATURAL-V.P.A.M (Visually Perfect Algebraic Method), Sharp’s WriteView, TI’s MultiView
Some Calulators that uses it: Casio fx-991ES (115ES|U.S), Casio fx-991ES PLUS, Sharp EL-W506, TI-30XS MultiView
Reverse Polish Notation (RPN)
- Traditional RPN (4 level stack)
Description:The traditional RPN first appeared on the HP-35, the world’s first scientific pocket calculator, and is still now used is some HP calculators, like the hp 35s. To add 2+2, you press:  [ENTER]  [+]. The 4 level stack are named x, y, z, t. With the x and y register showing on dual line display and only the x register in the case of a single line display calculator. Values entered are directly entered on the x register and when [ENTER] key is pressed, the value is copied to the y register. In modern calculators, only HP still offers RPN on their calculators.
Some Calculators that uses it: HP-35, HP 35s, HP-12C, HP-42S
- Reverse Polish LISP (RPL) input method
Description:First appeared on the HP-28C. To add 2+2, you press:  [ENTER]  [+]. Unlimited stack, only limited by memory. A few lines of the stack (more than traditional RPN) is shown on the dot matrix screens of these calculators, 7 on the HP 50g. Values entered are entered on a “temporary holding area”, not on the x register as in traditional RPN, when [ENTER] is pressed, the value is copied to the x register, but not to the y register. Used in HP series of graphing calculators.
Some Calculators that uses it: HP 28/48/49 series of calculators
RPN and RPL. Worth a read if you want to know more about RPN.