The Ottoman Empire, as with many powers of the period that lacked a first-rate native armaments industry, contracted much of their firearms from Mauser in Germany. They went through several contracts during the late 19th century, following along the trajectory of general Mauser development in that period, as the Ottomans were big fans, watching each new model closely and wanting to stay current with their models. You'd be hard pressed to find a customer for Mauser exports who cycled through more models in that period than the Ottomans! The irony of this of course was that these fancy modern rifles are something of a puzzle for many of the rural peasants who stocked the ranks of the Ottoman Army. German observers thought quite poorly of them, and even Ottoman officers would note during the Balkan Wars that:
Most of the men were conscripts who had never undergone modern training, and some of them did not even know to load a rifle properly. Rifles were handed out in the field directly from their packing crates, I even saw some soldiers trying to load bullets down the muzzle of the gun!
The Ottomans were very good at buying rifles, not good at handing them out, you might say, with many arms just sitting in storehouses until needed for conflict. The situation was perhaps slightly improved by WWI, but barely, and training still was incredibly lackluster, if happening at all. And in any case, it wasn't like the Ottomans were going to go back to the muzzleloaders that might have been more comfortable for the conscripts anyways.
Now, as for what they carried...
Mauser 1903: This was the primary arm of the Ottoman Army in the period, and little different from the venerable Gewehr 1898 that their German allies carried as well, with what minor changes that did happen being unnoticable to the casual observer. It was chambered for the 7.65x53mm round, which was the standard ammunition of the Ottomans, and many other buys of export models such as the Belgians and Argentines, although not the Germans themselves, whose G98s chambered 7.92×57mm, also known as '8mm Mauser'. However, there were nowhere near enough 1903s to go around, so despite being the premier arm of the Ottomans, it was hardly the only one.
Model 1905 Carbine: Putting this second since although a newer model, it was a minor one, with only 20,000 or so made for the purpose of arming cavalry and artillery troops. As with most other Turkish Mausers, it used the 7.65x53mm round.
Mauser 1893: The 1893 pattern was one of the major 'stepping stones' in Mauser development, introducing the flush-magazine which was facilitated by the staggered arrangement of the rounds. In addition to the Ottomans, the Spanish were the other major buyer, hence you'll also find it referred to as the "Spanish Mauser". Despite the newer 1903, the 1893 was still held in great numbers - just over 200,000 had been ordered - and commonly issues to frontline troops during the war, which wasn't too problematic as it chambered the same 7.65x53mm as the 1903.
Mauser 1890: This was basically the Belgian 1889 model, and although a bit dated compared to the newer models, and missing some of the newer improvements to the Mauser 'line', such as still having that jutting, single-stack magazine, it was still a perfectly serviceable rifle chambering that same 7.65x53mm round, which allowed it to be easily integrated into the logistical chain, and thus issueable for frontline use. A Carbine version was also made in very limited numbers
The Mauser 1887: This was a blackpowder rifle firing a big 9.5x60mm round, and long out of date. A variation of the Model 1871/84 in German service, the Ottomans contracted for over several hundred thousand of them in 1887, so as you might suspect, still had quite a bundle lying around. Beggars can't be choosers though, so even though it was long replaced as the principal arm of the Ottoman military, it continued to be issued in the war primarily for non-frontline troops such as guards and trainees (insofar as the latter even existed).
Peabody-Martini 1874: Prior to their 'love-affair' with Mauser, the Ottomans were mainly buying from the US, being the biggest importer of American-made arms in the period, and one of the last major arms contracts before the switch was via the Providence Tool Company for this single-shot rifle (actually a Martini-Henry, naming conventions is mostly revolving around patent disputes, see article below if this interests you) back in the early 1870s. Stores remained into the 20th century, and despite being long out of date, remaining ones were updated from blackpower 11.43x55mmR to fire the 7.65x53mm. These are refered to as Model 1874/12 or 1874/14 in some places, but the literature is so scant I'm not sure if that was an official designation or one collectors have applied in hindsight!
Handguns: Turkish officers didn't always carry handguns, the armament apparently just not being as popular with them as in most other militaries, so numbers are pretty small, all in all. Many officers who did want one would purchase privately, in some cases the same as 'issued' models, but you could find a real hodgepodge. Of note, though at least as far as official purchases go, notable ones include:
FN-Browning M1903: Several thousand of these pistols were purchased in the early 20th century and saw use both for the military and the police. The earlier Model 1900 was also present in some number
Mauser C96: Even for their handguns, the Ottomans looked into Mausers, although less exclusively (above). Although they didn't take them in great numbers - buying only 1000, they were one of the first foreign buyers, placing their order in 1897, and so nicknamed the "Cone-hammer" Mauser, as it was one of the few with that design which was soon changed.
Smith and Wesson No. 3 Revolver: Purchased in several successive contracts in the 1870s, but still to be found as late as WWI, as with the C96, these revolvers were bought in comparatively smaller numbers than similar contracts, several batches which rarely numbered more than 1000, which compares to say, the Russians who bought several hundred thousand of these through the years.
Sources
Achtermeier, William O. "The Turkish Connection: The Saga of the Peabody-Martini Rifle". Man At Arms Magazine, Vol. 1, No. 2, 1979. pp. 12-21
Ball, Robert W.D. Mauser Military Rifles of the World. Gun Digest Books, 2011. This is THE standard reference for Mauser rifles.
Grant, Jonathan A. Rulers, Guns, and Money: The Global Arms Trade in the Age of Imperialism. Harvard University Press, 2007.
Grüßhaber, Gerhard. The German Spirit in the Ottoman and Turkish Army, 1908-1938: A History of Military Knowledge Transfer Walter de Gruyter GmbH & Co KG, 2018.
Mowbray, Stuart C. Bolt Action Military Rifles of the World. Andrew Mowbray Publishers, Inc., 2009.
Yorulmaz, Naci. Arming the Sultan: German Arms Trade and Personal Diplomacy in the Ottoman Empire. I.B.Tauris, 2014.
I'd also recommend /u/othais' C&Rsenal "Primer" series on YouTube if you want good visual accompaniment, although I think they have only covered the Model 1887 so far when it comes to Ottoman arms. If you just want pretty pictures though, check out his site which has most of these.