Africa does not have tall, jagged mountain ranges comparable to the Himalayas, Andes, Rockies, or Alps and there are very few mountains requiring technical gear. The Atlas Mountains across Morocco, Algeria, & Tunisia; the Drakensberg in South Africa & Lesotho; the Semian Mountains in Ethiopia; and the Rwenzori Mountains between Uganda & the DRC are the only considerable mountain ranges on the continent, all with numerous peaks which can be easily climbed. Additionally, there are some tall volcanoes along the Great Rift Valley, on the Indian Ocean islands, & in Cameroon. Some of the continent's most climbed or unique mountains are:
Mount Kilimanjaro (5895m) in Tanzania near the Kenya border is the continent's highest peak, the world's tallest free-standing mountain, and perhaps the most climbed mountain on the continent, owing to its accessibility and the lack of need of technical gear. The range of scenery one passes from base to peak makes it a destination almost all climbers have on their wishlist.
Jbel Toubkal (4165m) near Marrakech, Morocco is the tallest peak in the Atlas Mountains and can be climbed without technical gear in summer.
Mount Cameroon (4040m) in Cameroon is an active volcano that rises straight out of the ocean and is covered in tropical forest and almost always shrouded in clouds/mist. Fast-paced hikes to the top and back are possible in a day.
Mount Nyiragongo (3470m) in the DRC on the Rwanda border is one of just 3-4 volcanoes in the world with a lava lake in its crater. A climb takes ~8 hours and involves camping on a ledge at the top?a safe 700m above the lake?for the night (of course, the steaming, bubbling lava is more spectacular at night).