Janissaries are extremely effective against any pre-Renaissance unit. For this reason, it's recommended to get Gunpowder as soon as possible, before any opponents so you have the upper hand against anything they send at you.
The bonus against Mounted units ensures your lands are mostly safe from pillaging, and any fast response to your Janissaries will be compromised. The bonus against Melee units helps defend your cities, and deepens the divide between Maceman and Musketman strength. Finally, the bonus against Archery units makes the Janissaries highly effective attacking units.
Despite the Janissaries' +25% against Mounted units, they are slightly less effective than Pikemen against mounted units. Knights may be ideal to attack them. Once you research Gunpowder, the Ottomans lose their advantage - promote Musketmen in Pinch and you can stop their upper hand.
Additionally, as the Ottomans will probably rush towards Gunpowder, they'll have missed out some other key technologies along the way and are likely to be picking them up - with a war-torn economy. There's a fair chance you can beat them to Riflemen.
The Janissaries were an elite fighting force of infantry raised by the Ottoman sultans. They formed the sultan's household troops and bodyguard, eventually rising to a position of enormous influence before being disbanded in the nineteenth century. Murad I founded the Janissary corps around 1330; the troops were made up of non-Muslim boys who were captured in battle or bought as slaves and brought up as warriors. This was very similar to the Mameluk slave soldiers of Egypt, who may have been the inspiration for the Janissaries. Janissaries became the first infantry units in the Ottoman military forces, because at the time no free man would consent to fighting on foot.
After 1380 the system for recruiting new soldiers into the Janissaries became standardized through a process called the "devshirmeh:" this was basically a "human tax" where each community had to contribute a certain number of young boys to the state. In practice these were almost always Christian slaves.
The boys were sequestered in special barracks, where they would be converted to Islam and trained to become elite soldiers (a process very similar to that practiced by the Spartans of ancient Greece). Although slaves, the Janissaries received special privileges, including the right to bear weapons and higher pay than other soldiers. Although the Janissaries began their existence as a small bodyguard for the sultan, in time they came to comprise much of the Ottoman army, topping out at over 200,000 men at some points.
The Janissaries originally were trained to be expert archers, but once firearms were invented they quickly adopted the new weapons, becoming fiercely proficient with muskets. They usually used axes or sabres for melee combat. Janissaries were present at all of the major battles in the early history of the Ottoman Empire, including the Siege of Constantinople in 1453.
The very success of the Janissaries would eventually destroy them. Realizing their importance, they began demanding higher pay, threatening rebellion if their demands were refused. Their first rebellion took place in 1449, and it was followed by many more. By the eighteenth century, the Janissaries dominated the government. Through frequent palace coups they changed sultans as they wished, and they blocked much-needed modernization reforms in the Ottoman army. Many of the Janissaries became corrupted, no longer training for battle but instead pursuing commercial ventures or political office, and the weakness of the Ottomans in the face of Russian and Austrian expansion soon became obvious to all. Desperate to regain control of their country, the Ottoman rulers made determined efforts to eradicate them.
Finally, in 1826 Mahmud II was able to form a non-Janissary army strong enough to challenge the Janissaries. The Janissaries did not go down peacefully, and Mahmud II had to use artillery to shell the Janissaries' barracks and batter them into submission. This event - known as the "Auspicious Incident," ended the Janissaries. Their characteristic marching music is the only trace of them that remains today.