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Assuming they are all the same grade (let's assume professional grade to simplify the question), what makes one better for one purpose or another? I understand that there are a variety to choose from: mars black, lamp black, chromatic black, etc.
Could someone tell me what the attributes of the different blacks are? (transparency, warmth, typical uses, etc)
The answer would be too broad because the different black pigments have all different characteristics as other pigments have in terms of hue/temperature, intensity/tinting, transparency, even value and so on. Here's what Winsor & Newton says about their acrylic blacks:
Ivory Black is a brown black of moderate tinting strength recommended for general use.
Mars Black is a denser, more neutral black with stronger tinting power.
Lamp Black is a blue black that is lightfast, permanent and opaque.
same (more or less) applies to oils, thus for a detailed description it's better to focus on pigment(s) rather than names as they are not universal between manufacturers. This applies to other hues more severely. For much accurate details you can refer to Handprint. And a chromatic black behaves as the pigments it's made of, so you can have more control over its attributes knowing what you've mixed.
There are two categories of blacks: carbon-based, and iron oxide-based.
They have different general characteristics: the single iron-based one, Mars black, is opaque and fast-drying, whereas the carbon-based pigments - most other black pigments - are slow-drying (unless drying agents are added during the production process), and more transparent.
Lamp black (carbon -, smoke -, soot -, oil -, blue -, Germantown -, flame black / blacking / Pigment Black 6, PBk6)
A blueish black. The fattest of the mentioned blacks.
Originally made from the soot that would accumulate on lamps, and currently still made using pure carbon: especially the more traditional recipes might contain impurities that increase drying time, so umber often was used as a drying agent. After charcoal black, it has the highest transparency.
Carbon black (Pigment Black 7, PBk7)
"Today, PBk7 is made today by burning coal or natural gas, but is traditionally derived from carbonising plant matter, like in the production of charcoal. Paints made with PBk7 are usually very opaque and have a high tinting strength thanks to the small, dense pigment particles. PBk7 can range from neutral to cool in temperature."*
I found this to be the most neutral of the blacks I've tried to find (i.e. PBk7—not "carbon black" (see note below)).
Vine black (charcoal -, yeast -, grape -, kernel -, drop -, cork -, mare -, Spanish -, German -, Frankfort -, blue black / Pigment Black 8, PBk8)
A pigment made from pyrolyzed vegetable sources (originally grapevines), but with lower intensity and tinting strength than ivory or lamp black.
Charcoal black (peach -, vegetable -, vine -, birch -, blue -, soft -, willow -, beech black / Pigment Black 8, PBk8)
Light, most transparent of blacks, brownish black, with very low tinting strength.
High transparency and low density are probably due to the relatively large particles (a personal guess). Best charcoal blacks are made from even-textured wood species. It is known that Rubens used charcoal to colour his imprimaturas grey. Note that the name is also used as an umbrella-term for all carbon-based blacks. I name this pigment in addition to 'vine black' as it is available as a high-transparency alternative.
Ivory black (bone -, Paris -, Frankfort -, German black / abaiser / Pigment Black 9, PBk9)
A brownish, (semi-)transparent, high-density black.
Has the most immoral history of the blacks, but is nowadays produced from the bones of 'regular' animals (mixed with Prussian blue). Most common black, and best suited for general use. Note that the traditional recipe is blueish with a yellow undertone, begging the question if the contemporary one is actually mainly carbon-based.
Mars black (black iron oxide / magnetic oxide / Mapico black / Pigment Black 11, PBk11)
Opaque, fast-drying, slightly warm but most neutral of the common blacks, with high drying agency, and really high tinting strength (best among its colour). Leanest of the blacks.
Named after a 18th century company (named after the Greek god of war); now synonymous for synthetic iron oxide pigments (compare e.g. mars red).
All aforementioned pigments have excellent light-fastness.
Chromatic blacks are, as the name implies, non-black pigment-containing mixes, that usually combine two complimentary colours. There are several kinds, most likely differing per manufacturer in transparency, temperature, tone, density, &c. I have no experience with these, but can imagine you can find a chromatic black for most specific purposes.
Deep blacks can also be mixed using transparent hues. A beautifully rich, transparent, dark black can be obtained by mixing either ultramarine blue with burnt umber, or quinacridone rose with emerald green or a blue like Prussian, phtalo, or ultramarine (which are also great for glazing). An advantage of this is that it's relatively easy to bring the mix to a warmer or cooler black.
As you probably immediately noticed, naming conventions are a mess. Because of the long history of pigment creation traditions and methods, monikers often are shared among distinct pigments. The most accurate way to talk about pigments is to refer to them by their Colour Index Generic Name (CIGN) Code.
Changing recipes also mean variable characteristics, so the information in this list unfortunately should be taken with a grain of salt. In this sense, user174174 is right in their answer as well: to get specific information, it's best to consult the web or manufacturer's website for the specific series and pigment.
That being said, this overview might serve as a guideline.
* https://www.jacksonsart.com/blog/2020/11/27/exploring-the-differences-between-black-pigments/
