Ttdfjt

Here is a discussion of fragile command in LaTeX. And it says that any commands that have an optional argument are fragile, for instance footnote[2]myfoottext and footnotmyotherfoottext. I have tried the example below which the commmand b has an optional parameter but it works without protect.

documentclassarticle
defb[#1]#2.#2.bf #1
begindocument
tableofcontents
sectionprotectb[one]two %works
sectionb[one]two %also works
enddocument

Here is the code in the .toc file:

contentsline sectionnumberline 1b [one]two1% 
contentsline sectionnumberline 2.t.bf onewo1%

However, it shows the effect of the command protect .

The command you defined does not take an optional argument, it takes a delimited argument. If you do:

defb[#1]#2.#2.bf #1
b[one]two

it will work fine, however if you remove the [one] TeX will throw an error:

defb[#1]#2.#2.bf #1
b two

! Use of b doesn't match its definition.
l.5 b t
 wo
?

because when you define a command with defb[#1]#2.#2.bf #1, TeX expects that when you use b, the input matches exactly the parameter text (i.e., [#1]#2), which means that the next token must be [, and when it is not that error is raised. See here for a brief description of that.

When using just def none of the arguments are optional! However, let's say you define:

newcommandb[2][--empty--].#2.bf #1

then the command will have 2 arguments, the first of which is optional, and if not given the default value is --empty--. When you use b, the defined command does not actually take any argument, but it checks if the next character is a [. If it is, the command proceeds to use an "inner" b (let's call it b@opt), which is defined as you did, with defb@opt[#1]#2.#2.bf #1. However if you use b without the following [, then a b@noopt is used, which is defined as defb@nooptb@opt[--empty--]. So after all you end up using b@opt, but the underlying definition provides the optional argument if you don't give it one.

You can manually define that with:

makeatletter
defb%
 @ifnextchar[%
 b@optb@noopt%

defb@nooptb@opt[--empty--]
defb@opt[#1]#2.#2.bf #1
makeatother

Now, what makes the optional argument thing “fragile”?

A command is fragile when it can't work properly in an expansion-only context, which is usually when being written to a temporary file, like in section headings as you showed, captions, and the like, but also inside an edef or, more recently, in expanded.

It's said that commands with optional arguments are fragile because the mechanism that checks if an optional argument is there (precisely, the @ifnextchar macro above) usually is fragile. It is possible, under some restrictions, to check for an optional argument expandably, like in xparse's NewExpandableDocumentCommand, but usually that's not the case.

Taking the command defined above as example, if you do edeftestb[one]two (or write or expanded) TeX starts expanding from left to right, so the first thing it sees is b, which is expanded to

@ifnextchar[b@optb@noopt

Next the @ifnextchar test is expanded to:

letreserved@d=[%
defreserved@ab@opt%
defreserved@bb@noopt%
futurelet@let@token@ifnch

Here the problem appears. let, def, and futurelet aren't expandable, so TeX leaves them as they are, and proceeds expanding the rest. All other macros there will be expanded by TeX, but by doing so the let and def will not define reserved@d and such, but their expansion, and this will make the code not work as intended.

Of course this is just an example, but the basic principle of fragility is that a command that contains non-expandable tokens is being used in an expansion-only context.

How to make a command robust?

Until a couple decades ago the only way to make a command robust was to prevent its expansion with noexpandcommand, which makes TeX temporarily treat command as unexpandable and skip it in an expansion-only context. The downside of this is that as soon as the expansion was carried out the noexpand would disappear and the command would be fragile again.

To circumvent this LaTeX defines protect and the accompanying macros protected@edef and protected@write, which define protect as defprotectnoexpandprotectnoexpand. Then, in an expansion-only context protectcommand will expand to noexpandprotectnoexpandcommand. TeX will throw both noexpands away, temporarily making protectcommand both unexpandable. If you happened to use the command again, it would continue being robust if you used the protected@... macros instead of the normal ones.

Commands with optional arguments defined with LaTeX2ε's newcommand and the like have a different look (but the same machinery underneath). If you define newcommandb[2][--empty--].#2.bf #1, then b will actually be protected@testopt b \b --empty-- (that \b is the command \b, with two backslashes, not \ then b). protected@testopt will use the protect machinery to test whether it can be safely expanded. If it cannot it will leave protectb, otherwise it will proceed to use \b, which contains the actual definition of the command.

All this became easier when ε-TeX introduced the protected primitive, which allows you to make a macro engine-protected. This means that instead of tricking TeX to noexpand your macro, you will define the macro as robust with:

protecteddefb%
 @ifnextchar[%
 b@optb@noopt%

and then TeX itself will know that b is not supposed to be expanded inside an edef or write or expanded, without additional machinery.

LaTeX2ε doesn't use protected to define robust macros because of backwards compatibility. LaTeX2ε predates ε-TeX, so the protection mechanism was established much earlier. LaTeX3, for example, dropped the 2ε protection mechanism and uses only protected to define robust macros.

As a side note, I'd change that definition of yours to:

newcommandmybold[2][--empty--].#2.textbf#1

and use as:

mybold[one]two

I changed the command to mybold, as one-letter command names are not generally a good idea. I also changed bf (which is deprecated for decades now) to textbf and put the second argument in braces, so that the second argument is two, not just t.

The information on that page is wrong (or at least outdated, all commands in latex2.09 that had an optional argument were fragile, but latex2e has been available since 1993...)

the example in the question does not define an optional argument but if you change it so that it does, using the facility of newcommand to define such an argument you will see that the resulting command is robust and this works without error

documentclassarticle
newcommandzb[2][?].#2. textbf#1
begindocument
tableofcontents
sectionzb[one]two %works
zzz
sectionzbthree %also works
zzz
enddocument

If you look at the .toc file you will see that this did not "blow up" the way a fragile command would, it produces

contentsline sectionnumberline 1zb [one]two1% 
contentsline sectionnumberline 2zb three1% 

Latex defines zb here in such a way that the protect mechanism is used internally so you do not need to explicitly use protect, so such command are, by definition, robust.

Of the list in that page

All commands that have an optional argument are fragile.

As noted above any commands with optional argument defined by newcommand (as well as some others) are robust (this has always been the case in LaTeX2e)

Environments delimited by begin ... end are fragile.

yes (we might fix that one day)

Display math environment delimited by [ ... ]

No, [ has been robust since the 2015 release.

Math environment ( ... )
However, $ ... $ is robust

No, ( has been robust since the 2015 release.

Line breaks, \

No, \has been robust since the 1994 release

item commands

Yes.

footnote commands

Yes.