Folliculitis: This is the most likely diagnosis given the location under the armpit, a common area for folliculitis due to friction, sweat, and bacterial growth. Folliculitis is an inflammation of the hair follicles, often caused by bacterial infection, and can present with redness, swelling, and pus.

Infected Lymph Node (Lymphadenitis): This could be a possible diagnosis if the patient has a recent history of infection or immune system compromise. Lymphadenitis presents with swollen, tender lymph nodes and can be caused by bacterial, viral, or fungal infections.
Hidradenitis Suppurativa: Although less common, hidradenitis suppurativa is a chronic skin condition that involves recurrent, painful abscesses and nodules in the apocrine-gland-bearing areas of the body, such as the armpits. It could be considered if the patient has a history of recurrent skin infections or abscesses in the same area.
Cellulitis: This is an infection of the skin and subcutaneous tissues, which could occur in the armpit area. It presents with redness, swelling, warmth, and sometimes fever.

Cat-Scratch Disease: Caused by Bartonella henselae, this infection can lead to lymphadenopathy, typically in the axillary region if the scratch is on the arm. It's crucial to ask about exposure to cats.
Tuberculosis (TB): Although less common in some regions, TB can cause lymphadenitis, and it's critical to consider this diagnosis, especially in patients with risk factors for TB exposure or immunocompromised individuals.
Lymphoma: While rare, lymphoma can present with lymphadenopathy. A thorough history, examination, and potentially further diagnostic tests are necessary to rule out this serious condition.

Actinomycosis: A rare bacterial infection that can cause abscesses or lymphadenitis, often related to poor dental hygiene, trauma, or immunocompromised states.
Sporotrichosis: A fungal infection that can cause lymphocutaneous lesions, typically acquired through skin trauma and exposure to contaminated soil or plants.
Atypical Mycobacterial Infections: These can cause lymphadenitis or skin infections and are more common in immunocompromised patients or those with specific exposures (e.g., swimming pools, aquariums).

The initial approach should include:

A thorough history and physical examination to differentiate between folliculitis, infected lymph nodes, and other conditions.
Laboratory tests may include complete blood counts (CBC), blood cultures, and potentially a biopsy or aspiration of the affected area for culture and histopathological examination.
Imaging studies like ultrasound can help differentiate between lymphadenitis and other conditions.
Antibiotics may be started empirically for suspected bacterial infections, with adjustments based on culture results.
Consider referral to a specialist (dermatologist, infectious disease specialist) for further management, especially if the diagnosis is unclear or if the patient does not respond to initial treatment.