An anal-fixated person is someone who is excessively tidy and obsessed with neatness, appearance and order. Such people are bureaucratic-minded, petty-minded jobsworths who make other people's lives a misery by creating unnecessary hurdles, and/or who lead a sad life due to their incapacity to set aside their obsession with regularity. It is often used as an insult in cases where people are overly strict or rigid about something.
The origin is that Freud, in his early work, said that young children go through three stages, oral, anal and phallic, before the age of 5, and that successfully traversing each stage is crucial to later psychological wellbeing. (The phallic stage produces the Oedipus Complex or in girls, the Electra Complex and penis envy). If someone fails to traverse a stage, usually because of excessively rigid parenting (though also sometimes through the opposite), they become "fixated" and have unresolved libidinal (i.e. psycho-sexual) issues which overshadow their later lives.
The anal stage occurs around the time a young child is potty-trained, and can include experiencing pleasure from either retaining or releasing shit. The characteristics usually called 'anal-fixated' are those of the anal-retentive, who gained disproportionate enjoyment from holding shit in, possibly because of too-rigid potty training. (An anal-fixated person can also strictly speaking be anal-expulsive, which leads to characteristics such as messiness etc., but this has not passed over into popular use). It expresses itself in later life in neuroses centred on rigidity and regularity.
In popular usage, usually abbreviated to "anal".
I'm only late because some anal-fixated shop assistant decided my credit card signature needed going over with a fine tooth-comb.