What is the lumpenproletariat?
First off, the lumpenproletariat are proletariat. They are part of the working class ("owning nothing but their labor"). In general Marx broke society into two high-level classes: bourgeoisie and proletariat. They are defined by their relationship to the means of production (bourgeoisie own the means of production, while the proletariat own only their own labor).
Eventually, he would clarify other marginal groups (such as the petty bourgeois you already mentioned). The lumpenproletariat are a subset of proletariat. They do not own the means of production. However, while the ordinary proletariat were constructed to be participants in the capitalist system (by selling their labor to the bourgeoisie), the lumpenproletariat do not participate. The lumpenproletariat do not consistently sell as other proletariat do.
This is not an entirely discrete category, but I'm not sure if that is because Marx never fully specifies it or because it is naturally fuzzy. Certainly some members of the lumpenproletariat do sell their labor (porters, organ grinders, prostitutes), but my reading is that they are distinguished from ordinary proletariat because of their intense marginalization. Other workers are somewhat marginalized, but only the lumpenproletariat are completely marginalized from society. That's what the "lumpen-" part means, they are "rags" - worthless parts.
Why are they important?
Because they have a different strategic role in the glorious workers' revolution.
Marx thought that the lumpenproletariat - unlike the general proletariat - would not be a valuable part of the revolution. On one hand, their life experience means they could be excellent revolutionaries. Unfortunately, they are easily swayed by money. Having nothing, their services can be easily bought by the anti-revolutionary factions.
This is described in the 18th Brumaire, including the section you quoted in the question. In that section, Marx describes how the lumpenproletariat had organized into secret societies (a good revolutionary tactic), but that Bonaparte had placed his people in charge of the secret sections (coopting revolutionary workers for his ostensibly bourgeois organization).
This role is also mentioned in the Communist Manifesto, where they have been called the "dangerous class". Here it is mentioned that they have a reactionary bent: since outcasts and criminals leech their support from the wealthy, they have a vested interest in keeping the wealthy in power.
Who are they today?
Fundamentally, the lumpenproletariat are unemployed members of the proletariat. Your question mentioned that criminals and welfare recipients are presumably members, but that isn't true. Many criminals have full-time jobs, some pay excellently (for example, white collar criminals). Many welfare recipients are also employed.
Both of those groups lack the second quality also - which is an intense marginalization.
Think more along the lines of the perennially homeless or vagrants, as well as the kinds of criminals who exist alongside harsh urban poverty.
This answer was greatly influenced by this page, written by NYU Professor of Politics Dr.Bertell Olmann.